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Black Jack |
How Lucky Are You?
Our last two assignments are statistical experiments. Statistics is all about comparing what we expect to get with what we actually get. What we expect to get is called the Expected Value. What we actually get is called the Observed Data. In our experiments, as scientists, we expect nothing significant to occur. Expecting nothing significant is the same as assumimg events will occur randomly. Random events are chancery, haphazard, and chaotic, without cause or purpose. Statisticians are always hoping their assumptions are wrong— that something unexpected will happen, like someone winning the lottery twice, rolling seven 7's in a row, or a new drug that cures everyone of AIDS. When it does, they know their experiment was not an accident, and something other than chance determined the outcome of events. In other words, they have discovered significant results. Significance is measured by finding the difference between the expected and the observed. If the difference between the two is big enough, then we conclude:
For more about statistics, there is always the Wikipedia.
Today you will find out if you are lucky by comparing probable results with actual results. First, you will compute what you would expect by chance. Second, you will collect data by playing the game of Blackjack 104 times. Third, you will measure the difference between the probable outcome and your actual data to see if you scored significantly better than chance expectations. If so, we will call you Lucky.
ASSIGNMENT
Below is an EXPECTED PROBABILITY table for all of you to fill in. Copy the table to a comment and answer the questions as you have done in previous assignments. When you answer a question, or change someone's answer, explain why. Below are some definitions to help you answer the questions.
DEFINITIONS:
- A DECK of cards is a set of 52 cards.
- The deck contains 4 subsets called SUITs.
- The 4 suits are called Diamonds, Clubs, Hearts, and Spades.
- Each card in a deck belongs to one and only one of these suits.
- Each card in each suit has a RANK from Ace to King.
- The 13 ranks are: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King.
- Thus, there are 4 cards of each rank, one for each suit.
- There are 4 Aces, one for each suit.
- Jacks, Queens, Kings are called FACE cards.
- There are 16 cards that are either a FACE card or a TEN.
- A BLACKJACK is two cards: one an ACE and the other a FACE card or a TEN.
The order does not matter. - You win each time you are dealt a hand of Blackjack.
- To DEAL a card is to take a card off the top of the deck.
- A DEAL in Blackjack is any two cards dealt off the top of the deck. The order matters.
- A HAND of Blackjack is any two cards dealt off the top of the deck. The order does not matter.
QUESTIONS:
- EXPECTED PROBABILITY:
- How many ways can you deal a Blackjack?
- How many Blackjack hands are there?
- How many ways can you deal 2 cards?
- How many 2 card hands are there?
- What is the probability of dealing a Blackjack? (round off 3 places)
- You should expect a Blackjack once in every ______ hands. (round off)
- How many Blackjacks should you expect in 104 hands?
Copy this table to a comment, and answer the questions.
If you disagree with someone, tell us why in a comment.
EXPECTED PROBABILITY
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | ||
| 2 | How many Blackjack hands are there? | ||
| 3 | How many ways can you deal 2 cards? | ||
| 4 | How many 2 card hands are there? | ||
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | ||
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | ||
| 7 | How many Blackjacks should you expect in 104 hands? |
Below is an OBSERVED DATA table for each of you to fill in.
Copy the table to a comment and fill it in with your OBSERVED DATA.
The OBSERVED DATA is collected by pushing the Shuffle & Deal button 4 times.
- BLACKJACK APPLET INSTRUCTIONS
- Set your monitor display to 1024x768 so you see the entire applet area.
- Click the [Shuffle & Deal] button to deal your deck of cards.
- The applet deals 2 cards at a time until it deals 26 hands.
- Blackjacks are indicated by BLACKJACK! .
- The applet counts the number of Blackjacks.
- Do not scroll up and down. The applet generates errors if you do.
- Repeat steps B to E three more times.
- When finished, you will have dealt 104 hands.
- The applet totals all your Blackjack hands from the 4 shuffles and deals.
This is the answer to question 8.
Do not scroll up and down between shuffles and deals, or your count will be wrong.
Do not cheat by continuing to play till you accumulate beaucoups of Blackjacks.
There is a maximum number possible.
And there is no point in it.
The OBSERVED DATA table asks these questions:
OBSERVED DATA:
8. How many Blackjacks did you deal in 104 hands?
9. What fraction of 104 is your number of Blackjacks?
10. What is the class average so far for the number of Blackjacks in 104 hands?
Compare the class average to question 7.
If one number is almost twice the other, rethink question 5.
COMPARE EXPECTED PROBABILITY WITH OBSERVED DATA:
11. What is the positive difference between line 7 and line 8?
12. Divide line 11 by line 7.
13. Write line 12 as a percentage.
This is the % error between the expected results and your observed data.
The % error determines how lucky you are.
If line 8 is more than line 7, you are lucky by the amount on line 13.
If line 13 is more than 99%, you will soon be banned from every casino.
If line 8 is less than line 7, you are unlucky by the amount on line 13.
If line 8 is 0, you are a cooler. They will love you in Vegas.
14. Are you lucky? If line 13 is more than 99%, enter YES, otherwise NO?
After you have pushed the Shuffle & Deal button 4 times,
copy the table below to a comment,
and enter your OBSERVED DATA.
Then answer the questions.
After you enter your data, enter the class average for column 8, so far, on the bottom line.
OBSERVED DATA
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| Average |
As a class, you may earn a group score of 7 points.
As individuals, you may earn another 7 points.
But you will not earn any points till you do the experiment!
That's a big sum of 14 points for this assignment!
Comments:
From wHolt - 4/2/06 12:52 AM
Comments:
From wHolt - 12/20/06 12:58 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | ||
| 2 | How many Blackjack hands are there? | ||
| 3 | How many ways can you deal 2 cards? | ||
| 4 | How many 2 card hands are there? | ||
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | ||
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | ||
| 7 | How many Blackjacks should you expect in 104 hands? |
From TBird - 12/15/06 8:28 AM
| 1 | How many ways can you deal a Blackjack? | 128 | 4*16=64*2=128 |
| 2 | How many Blackjack hands are there? | 64 | 128/2=64 |
| 3 | How many ways can you deal 2 cards? | 2652 | 52*51=2652 |
| 4 | How many 2 card hands are there? | 1326 | (52*51)/2 if the order does not matter |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 4.827% | 64/1326=.04827 |
| 6 | You should expect a Blackjack about once in every hands. | 21 | |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 6 | 6/104=.057 | 1 | (6-5)/5= .2 | 20% | not really | |
| Average | 4 | 0 | 0 | 0 | 0 | no |
From Bubba - 12/14/06 4:01 PM
| 1 | How many ways can you deal a Blackjack? | 128 | 4*16=64*2=128 |
| 2 | How many Blackjack hands are there? | 64 | 128/2=64 |
| 3 | How many ways can you deal 2 cards? | 2652 | 52*51=2652 |
| 4 | How many 2 card hands are there? | 1326 | (52*51)/2 if the order does not matter |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 4.827% | 64/1326=.04827 |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | every 21 hands or so | |
| 7 | How many Blackjacks should you expect in 104 hands? | only about 5 |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 6 | 6/104=.057 | 1 | (6-5)/5= .2 | 20% | NOOOOO | |
| Average | 4 | 0 | 0 | 0 | 0 | no |
From CenterField - 12/13/06 7:10 PM
| Answers | Why? | ||
| 1 | How many ways can you deal a Blackjack? | 128 | 16*4*2 or (16*4)+(4*16) |
| 2 | How many Blackjack hands are there? | 64 | 128/2; number of deals divided by 2 to eliminate repetitions since order does not matter in hands. |
| 3 | How many ways can you deal 2 cards? | 2652 | 52*51 |
| 4 | How many 2 card hands are there? | 1326 | 2652/2; same explanation as #2 |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | .048 | 128/2652; possible ways to deal a blackjack divided by possible deals |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 21 | 1/.048 or .048(X)=1 |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | 104/21 = 4.95 |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 4 | .038 or 1/26 | 4-5 = -1 | (4-5)/5 = -1/5 | 20% | No | |
| Average | 4 (rounded down from 4.381) | (I recorded 21 answers, including mine, Professor Holt's, and Boki's individual numbers,but excluding Kathi's old answer) |
From TBird - 12/13/06 12:31 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | 4 Aces*16 Faces*2 different ways to deal each one. Order does not matter. |
| 2 | How many Blackjack hands are there? | 64 | 4 Aces*16 Faces... Order does matter. |
| 3 | How many ways can you deal 2 cards? | 2652 | 52 ways for the first card*51 ways for the second. Order does not matter. |
| 4 | How many 2 card hands are there? | 1326 | (52*51)/2 Order does matter. |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | .0483=4.83% | Total ways to deal a Blackjack hand/ Total ways to deal a two card hand. |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 21 | 1 Blackjack hand/ .0483 probability. |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | 104 hands* .0483 probability. |
From Bubba - 12/11/06 4:20 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 64 | four different aces combined with the 16 other face cards which could add up to 21 |
| 2 | How many Blackjack hands are there? | 4 | in a round of black jack you use each card only once until all of the cards are dealt. and with only 4 aces in the deck you could only have 4 chanced. yet if like in vegas you use multiple decks, most use six, you increase your chaces. |
| 3 | How many ways can you deal 2 cards? | 26 | 52 cards divided by 2 = 26 |
| 4 | How many 2 card hands are there? | 1326 | (52*51)/2 if the order does not matter |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | .0.024 | 4/52 * 16/52 = .024 |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | every 21 hands or so | |
| 7 | How many Blackjacks should you expect in 104 hands? | only about 5 |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 6 | 6/104 | 5 | (6-5)/5 | 20% | NOOOOO | |
| Average | 4 | 0 | 0 | 0 | 0 | no |
From Houdini - 12/11/06 2:14 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | 4 Aces*16 Faces*2 different ways to deal each one. Order does not matter. |
| 2 | How many Blackjack hands are there? | 64 | 4 Aces*16 Faces... Order does matter. |
| 3 | How many ways can you deal 2 cards? | 2652 | 52 ways for the first card*51 ways for the second. Order does not matter. |
| 4 | How many 2 card hands are there? | 1326 | (52*51)/2 Order does matter. |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | .0483=4.83% | Total ways to deal a Blackjack hand/ Total ways to deal a two card hand. |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 21 | 1 Blackjack hand/ .0483 probability. |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | 104 hands* .0483 probability. |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 7 | .065=6.5% | 2 | .4 | 40% | Nope | |
| Average | 4 | .038=3.8% | -1 | .2 | 20% | Hell No |
No Blackjack for me, but hey, there's still the free drinks!
From Fro - 12/5/06 8:05 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | 4: Aces |
| 2 | How many Blackjack hands are there? | 64 | 4: Aces 16: face cards or tens. 4 x 16 = 64 |
| 3 | How many ways can you deal 2 cards? | 2652 | 52*51 = 2652 |
| 4 | How many 2 card hands are there? | 1326 | 2652 / 2 because order doesn't matter in hands |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | .048 or 4.8% | 4 Aces |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 21 | 1 /.048 = 20.833 |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | 104 / 21 = 4.952 |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 5 | .048 | 5 - 5 | (5 - 5) / 5 | 0% | NO | |
| Average | 4.15 |
From Melewen - 12/4/06 5:21 PM
| QUESTIONS | Answers | Why? | |
| 1. | How many ways can you deal a Blackjack? | 128 | 4 Aces x 16 other cards (4 tens, 12 face cards) = 64. 64 x 2 (since you're calculating deals and order matters) is 128 |
| 2. | How many Blackjack hands are there? | 64 | Order doesn't matter in a hand, so you take the possible blackjack combinations, but don't multiply it by 2. |
| 3. | How many ways can you deal 2 cards? | 2652 | You count the possibilities for when order matters like we've been doing. 52x51 |
| 4. | How many 2 card hands are there? | 1326 | Since order doesn't matter here, you divide your previous number by 2. 2652/2 |
| 5. | What is the probability of dealing a Blackjack? (round off 3 places) | 4.827% | There are 128 ways to deal a blackjack, divided by the total number of ways to deal 2 cards (2652). 128/2652 |
| 6. | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 20 | One out of a whole: 1/.04827 = 20.7. You round down since you haven't completely gotten to 21 to be able to depend on it for a probability |
| 7. | How many Blackjacks should you expect in 104 hands? | 5.02008 | The probability to get a blackjack times 104 hands; .04827x104 |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
| #BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 5 | .04807 | |5-5| = 0 | |5-5|/5 = 0 | 0% | Not particularly lucky, but not doomed either! I should come out even. | |
| Average | 4 (I put Boki's down as 4) |
From wHolt - 12/4/06 9:27 AM
Pac - question #2 is actually asking
how many different blackjack hands are possible,
not how many blackjacks are possible
dealing one time thru the deck.
Hope that's clearer.
7Iron - #11 should be in integers, not probabilites.
Kathi - the denominator in #12 should be the expected number,
not your observed number 3.
I noticed that you and 7Iron differed on class average.
From Kathi - 12/4/06 6:27 AM
Observed data redone:
Q: | #8 | #9 | #11 | #12 | #13 | #14 |
| | #BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? |
| | 3 | 3/104 = .029 | 5-3 = 2 | (5-3)/3 = .666 | 67% | No |
| Average | class average is 4 blackjacks | | | | | |
From 7Iron - 12/3/06 6:08 PM
| | QUESTIONS | Answers | Why? |
| 1 | How many ways can you deal a Blackjack? | 128 | 16*4*2 |
| 2 | How many Blackjack hands are there? | 64 | 16*4 |
| 3 | How many ways can you deal 2 cards? | 2652 | 52*51 |
| 4 | How many 2 card hands are there? | 1326 | 2652/2 |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 4.827% | 128/2652 |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 20 | 1=x*0.04826 |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | 104*0.04826 |
Q: | #8 | #9 | #11 | #12 | #13 | #14 |
| | #BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? |
| | 5 | 5/104 | 0.04807-.04739 | (.0068)/5 | 0.136% | No |
| Average | 5 | | | | | |
From Pac - 12/3/06 3:40 PM
Right - there are 128 ways to deal a blackjack (that's #1), but there are only 4 blackjacks per deck because there are only four aces per deck (that's #2). I see the difference in language that you mention, which is demonstrated in the differences of answers in #1 and #2.
With the average part of it, did you want me to take out the anomolous result when figuring the average?
From wHolt - 12/2/06 1:09 PM
So Kathi did something that happens sometimes, and the applet went bonkers.
However, even though only 4 blackjacks are possible each time thru the deck,
there are more than 4 possible ways to deal a blackjack.
A subtle difference in language here, but the concept is easy:
AK and KA are two different blackjack deals.
List all the other possible arrangements of Aces and Faces or Tens.
Then you will have a complete list of all possible blackjack deals.
From Pac - 12/2/06 11:41 AM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | 4 Aces; 16 face cards = 16*4=64; then two ways to deal that = 64*2=128 |
| 2 | How many Blackjack hands are there? | 4 | There are only 4 Aces in a deck, meaning that there can be only 4 Blackjack hands |
| 3 | How many ways can you deal 2 cards? | 2,652 | 52 cards * 51 cards after the first card is dealt = 52*51=2652 ways |
| 4 | How many 2 card hands are there? | 1,326 | 2652 from #3 / 2 (2 cards in a hand) = 2652/2=1326 |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 0.024 or 2.367% | (4/52 probability for the ace) * (16/52 probability for face or 10 cards) = 0.076923*0.307692=0.023669 |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 42 | Probability of blackjack is 2.367% so 2.367 hands out of 100 hands is once in every 42 hands. (100 hands / 2.367 probability of blackjack) |
| 7 | How many Blackjacks should you expect in 104 hands? | 2 | 2.367 blackjack probability per 100 hands, adding 4 hands doesn't change much (answer rounded to whole number; decimal number would be 2.496 blackjacks in every 104 hands) |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 3 | 3/104 | |3-2|=1 | |(3-2)/2|=0.5 | 50% | Nope | |
| Average | 5 |
I'm confused as to how everyone's average is so low (<5), especially since Kathi got 17 blackjacks. I included Mr. Holt's average and Boki's individual numbers and got 5.
From Trixie - 12/2/06 9:07 AM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | 4*16 = 64*2=128 |
| 2 | How many Blackjack hands are there? | 64 | 128 ways / 2 or 128 / 2 = 64 |
| 3 | How many ways can you deal 2 cards? | 2652 | 52 total ways for card 1, and 51 total ways for card 2. So, 52*51 = 2652. |
| 4 | How many 2 card hands are there? | 1326 | 2652/2 = 1326 |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 4.827% | 64/1326 = .04827 = 4.827% |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 21 | 1/ .04827 (prob. of getting a blackjack) = 20.716 = 21 |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | 104 (hands) * .04827 (Prob.) = 5.02 = 5 hands |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 2 | .019 | 2-5 = -3 | (2-5)/5 = .6 | 60% | No | |
| Average |
From wHolt - 12/1/06 1:38 PM
Remember deals do count order; it's hands that do not.
Use absolute value on #13. No -20% necessary.
Those of you who figured 1 in every 20.7 deals you would get a blackjack,
need to round off to the nearest whole number.
From Pringle - 12/1/06 9:19 AM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | ace can match with 4 differnet ranks(10,J,Q,K) and each rank has 4 suit. 4*4=16 and there are 4 ace 16*4=64 And order doesn't matter on blackjack, I double this. 16*2= 64 |
| 2 | How many Blackjack hands are there? | 128 | since order doesn't matter on blackjack(you just need those specific 2 cards) I think it's same as #1 |
| 3 | How many ways can you deal 2 cards? | 1326 | (52*51)/(2*1)= 1326 I got this fomula from amalgram assignment. n(n-1)..(n-k+1)/k! order does matter so has to be divided by k factor, which is 2 |
| 4 | How many 2 card hands are there? | 2652 | 52*51= 2652 order doesn't matter |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 0.097 | 128 / 1326 ( I used 1326 since it's dealing a blackjack)= 0.097 |
| 6 | You should expect a Blackjack about once in every ----------------- hands. | 10 | 1:0.097 = 1/0.097=10 |
| 7 | How many Blackjacks should you expect in 104 hands? | 10 | 104/10= 10.4 I expect to have blackjack once every ten time. So in 104 hands I should have 10 blackjacks. |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 6 | 0.058 | l6-10l=4 | l6-10l/10= 0.4 | 0.4*100=40% | no | |
| Average | 4.71 |
From Phoenix - 11/30/06 11:03 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | 4*16=64*2=128 |
| 2 | How many Blackjack hands are there? | 4 | There are only 4 Aces, so 4 sets |
| 3 | How many ways can you deal 2 cards? | 26 | 52/2 |
| 4 | How many 2 card hands are there? | 2652 | 52*51 |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 4.8% | 128/2652=.048 |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 20 | 1/.048 |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | 104*.048 |
From Zonino - 11/30/06 10:30 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | There are 4 each of Aces, Kings, Queens, Jacks, and Tens. In order to have a blackjack, you need 1 ace and 1 card with a value of ten. If we take the hands that start with an ace we would get 4*16 or the number of aces times face cards. This is 64. Logically, this is also the same number of ways to deal a hand that starts with a face card and the second card is an ace. We add 64+64 and get 128. |
| 2 | How many Blackjack hands are there? | 64 | We know there are 128 ways to deal the cards. For a Blackjack hand, it does not matter what order the face or ace comes out of the deck. Therefore we would divide by 2 to get the number of hands since order does not matter. This is 64. |
| 3 | How many ways can you deal 2 cards? | 2652 | There are 52 cards in a deck. You can deal the first card 52 ways and the second card 51 ways. 52*51=2652 |
| 4 | How many 2 card hands are there? | 1326 | Order does not matter in a hand. So we would take the number of ways you can deal a hand which is 2652 and divide by 2 since order does not matter. This is 1326. |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 4.827% | The probability is figured by finding out what the number of black jack hands are divided by the total number of hands. This is 64/1326 or 0.04827 or 4.827%. |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 20 | To find the number of whole hands we can use simple algebra = x*0.04827=1 blackjack x=1/0.04827 x= 20.7 or 20 whole hands. |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | We would take 104 hands * the probability of a blackjack (0.04827) = 5.02 or about 5 hands. |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 4 | 0.038 | |4-5| = 1 | |4-5|/5= 0.20 | 20 | NO | |
| Average | 4.5 |
From BassLady - 11/30/06 9:22 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | 1 suit of A's and 16 ways to deal a black jack and if you reverse the order that is 32. 32 * 4 = 132 |
| 2 | How many Blackjack hands are there? | 64 | 128 / 2 = 64 |
| 3 | How many ways can you deal 2 cards? | 2652 | 52 * 51 |
| 4 | How many 2 card hands are there? | 1326 | 2652 / 2 = 1326 |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 4.827 | 128/2652 |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 21 rounded | 1 / 4.827 |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | 104 / 21 |
From Capricorn - 11/30/06 9:07 PM
| | QUESTIONS | Answers | Why? |
| 1 | How many ways can you deal a Blackjack? | 128 | There are 16 face cards and ten cards. You multiply that by the number of aces, which is 4. Times the number of ways you can be dealt that hand which is 2. |
| 2 | How many Blackjack hands are there? | 64 | 16x4, you so nor multiply by two because order does not count. |
| 3 | How many ways can you deal 2 cards? | 2652 | 52 cards*51 cards(minus the one in use already)times 2(the number of ways it can be dealt) |
| 4 | How many 2 card hands are there? | 1326 | 52 cards times 51 cards and order does not count. |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 0.048 | 128 ways to deal the blackjack divided by the 2652 ways to deal two cards. |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 20 | 1 whole number hand / 0.048 the probability of getting blackjack. |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | 104 hands times the probability of getting a black jack 0.048= 5 blackjacks |
From Taurus - 11/30/06 8:58 PM
| | QUESTIONS | Answers | Why? |
| 1 | How many ways can you deal a Blackjack? | 128 | 16 cards that equal 10 times 4 aces times the 2 ways they can be dealt. |
| 2 | How many Blackjack hands are there? | 64 | 16 cards equal 10times 4 aces. |
| 3 | How many ways can you deal 2 cards? | 2652 | 52*51= 1326*2(order)=2652 |
| 4 | How many 2 card hands are there? | 1326 | 52*51=1326 |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 0.048 | 128/2652 (number of ways to deal blackjack/number of ways to deal two cards) |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 20 | 1/0.048 |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | 104*0.048=5.02 |
From Tiger - 11/30/06 8:52 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | 4*16=64*2=128 |
| 2 | How many Blackjack hands are there? | 64 | 4*16 |
| 3 | How many ways can you deal 2 cards? | 2652 | 52*51=2652 |
| 4 | How many 2 card hands are there? | 1326 | 2652/2=1326 |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | .048 | 64/1326 or 128/2652 |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 21 | 1/.048=20.8 |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | .048*104=4.99 |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 4 | 4/104=.038 | 4-5=-1 | (4-5)/5=-.2 | -20% | NO | |
| Average | 4.6 |
From Bubba - 11/30/06 8:24 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 64 | four different aces combined with the 16 other face cards which could add up to 21 |
| 2 | How many Blackjack hands are there? | 4 | in a round of black jack you use each card only once until all of the cards are dealt. and with only 4 aces in the deck you could only have 4 chanced. yet if like in vegas you use multiple decks, most use six, you increase your chaces. |
| 3 | How many ways can you deal 2 cards? | 26 | 52 cards divided by 2 = 26 |
| 4 | How many 2 card hands are there? | 2652 | 52*51 |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | .0.024 | 4/52 * 16/52 = .024 |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | every 26 hands or so | |
| 7 | How many Blackjacks should you expect in 104 hands? | only about 4 |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 4 | 1/26 | 0 | 0/4 | 0% | NOOOOO | |
| Average |
From Draco - 11/30/06 7:31 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | 16*4 |
| 2 | How many Blackjack hands are there? | 64 | (16*4)/2 |
| 3 | How many ways can you deal 2 cards? | 2652 | 52*51*50.../50*49*48... |
| 4 | How many 2 card hands are there? | 1326 | 52*51*50.../100*99*98... |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | .0483 or 4.83% | 128/2652 |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 20 | 1/.0483=20.7 |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | 104*.0483=5.0232 |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 4 | 4/104 | l 4-5 l=1 | 1/5=0.2 | 20% | NO | |
| Average | 4 |
From GolfGirl - 11/30/06 3:13 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | 4 x 16 = 64 hands. then 64 x 2= 128 |
| 2 | How many Blackjack hands are there? | 64 | 4 x 16 = 64 hands. |
| 3 | How many ways can you deal 2 cards? | 2652 | 52*51 = 2652 |
| 4 | How many 2 card hands are there? | 1326 | 2652 / 2 |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 4.827% | 128 /2652 = .04827 = 4.827% |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 20 | 1/.04827 = 20.7 |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | .04827 * 104 = 5.020 |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 4 | 4/104=.038 | 4-5=-1 | (4-5)/5=-.2 | -20% | NO | |
| Average | 5.2 |
From wHolt - 11/30/06 10:17 AM
Does counting order and not counting order ever result in the same number?
From Harkar - 11/29/06 8:02 PM
| | QUESTIONS | Answers | Why? |
| 1 | How many ways can you deal a Blackjack? | 128 | There are 16 cards that are either a FACE card or a TEN. There are 4 ACES. So 4 x 16 = 64 hands. Each blackjack hand can be dealt in two orders: ACE then FACE or FACE then ACE, so 64 x 2= 128 |
| 2 | How many Blackjack hands are there? | 64 | Again, there are 16 cards that are either a FACE card or a TEN. There are 4 ACES. So 4 x 16 = 64 hands. |
| 3 | How many ways can you deal 2 cards? | 2652 | 52*51 = 2652 (Order counts) |
| 4 | How many 2 card hands are there? | 1326 | 2652 / 2 (Order doesn’t count) |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 4.827% | 128 ways to deal a blackjack /2652 ways to deal 2 cards in a deck of 52 = .04827 = 4.827% |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 20 | 1/.04827 = 20.7 20 whole hands |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | .04827 * 104 hands = 5.020 |
OBSERVED DATA
Q: | #8 | #9 | #11 | #12 | #13 | #14 |
| | #BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? |
| 7 | 0.0673 | |7-5|=2 | |7-5|/5 = 0.4 | 40% | No | |
| Average | 3.625 | | | | | |
From wHolt - 11/29/06 1:30 PM
Boki - .483 means you expect a blackjack about half the time.
You are very optimistic!
Kathi - you did not get 17 blackjacks. That's impossible.
From Kathi - 11/29/06 10:07 AM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 128 | 4 aces and 16 face cards and tens. |
| 2 | How many Blackjack hands are there? | 64 | 4 aces and 16 face cards and tens. 4 * 16 = 64 |
| 3 | How many ways can you deal 2 cards? | 2652 | 52*51 = 2652 |
| 4 | How many 2 card hands are there? | 1326 | 2652 / 2 (ways to deal) |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | .048 | 4 ways to get an ace |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 20 | 1/.048 = 20.833 |
| 7 | How many Blackjacks should you expect in 104 hands? | 5 | .048 * 104 = 4.992 or 5 |
| Q: | #8 | #9 | #11 | #12 | #13 | #14 |
#BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? | |
| 17 | .163 | |5-17|=12 | |5-17|/5 = 2.4 | 240 | yes | |
| Average | 4.25 |
From wHolt - 11/27/06 10:09 AM
Boki -
how can you have more blackjack hands than 2 card hands?
Remember - deals count order; hands are sets, so they do not.
Also there is no way to get 512 blackjacks if you only deal 104 times.
Think how many blackjacks are possible each time thru a deck of cards?
Redo 8 thru 14.
Sometimes the counting does go crazy on this applet.
If so just redo the experiment.
From Boki - 11/26/06 11:30 AM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 64 | The deck consists of 52 cards subdivided into 4 suits because , and in each suit there are 13 cards (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King) there are 4 ways to get an ace, and there are 16 ways to get a face card or 10 |
| 2 | How many Blackjack hands are there? | 128 | 128 ways that a blackjack hand can be formed Since order doesn't matter, there will be 2*4*16=128 blackjack hands. |
| 3 | How many ways can you deal 2 cards? | 2652 | the number of ways to get any two cards from the 52 cards is: 52*51= 2652 |
| 4 | How many 2 card hands are there? | 1326 | 52*51/2*1= 1326 (identical 2 card hands) |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 0.241 | The probability of receiving a blackjack is 4*16/ (52*51)= 64/2652=0.241 |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | once in 21 hands | The probability of receiving a blackjack is 0.0483 or 4.83% If we divide 1 with 0.0483, we will get the number of hands: 1/.0483=20.703933 |
| 7 | How many Blackjacks should you expect in 104 hands? | By my observation: 4.139 Depends on how lucky we are; sometimes we can get 0, sometimes 1, sometimes 2, but sometimes several in a row. | The probability of receiving any blackjack once in every WHOLE NUMBER hands is 0.0483 (1 hand in 20.7 hands), but for any suited blackjack is .0483/4= 0.012 (1 hand in 83.33 hands) or 1.2%. We can also assume that is possible to get Blackjack from any of 2652 ways can you deal 2 cards, and in this case the probability will be 64/2652=.024 which is 2.4% an average chance to get a Blackjack, and number of hands will be 1/.024=41.66=42 (1 hand in 42 hands). I realized that either of above is correct because you could receive two blackjacks in a row and then receiving none over the next more than 84 hands. |
OBSERVED DATA:
Q: | #8 | #9 | #11 | #12 | #13 | #14 |
| | #BJs | BJs/104 | Obs-Exp | (Obs-Exp)/Exp | % | Lucky? |
| | 5 | 0.48 | |4.139-5|= 0.861 | 0.861/5 = 0.208 | 20.8 | NO |
| 4 | .038 | |4.139-4 |=0. 139 | 0. 139/4 = .035 | 3.5 | NO | |
| 2 | .019 | |4.139-2|=2.139 | 2.139/2 =1.07 | 107 | YES | |
| 4 | .038 | |4.139-4 |=0. 139 | 0. 139/4 = .035 | 3.5 | NO | |
| Average | 3 | 0.158 | |4.139-3|=1. 139 | 0.8195 /3 = .27 | 27 | NO |
From wHolt - 11/25/06 12:45 AM
TBird - Hands and Deals are not counted the same way.
In one order matters; in the other, order does not.
From TBird - 11/24/06 4:54 PM
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 64 | Because (4)(16) = 64 |
| 2 | How many Blackjack hands are there? | 64 | Because there are 64 ways you can deal a Blackjack |
| 3 | How many ways can you deal 2 cards? | 2652 | (52)(51) = 2652 |
| 4 | How many 2 card hands are there? | 2652 | Because there are 2652 ways to deal 2 cards. |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 0.024 | 64/2652 = 16/663 = 0.0241 |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 41 | 663/16 = 41.4375 |
| 7 | How many Blackjacks should you expect in 104 hands? | 3 | 104/(663/16) = 2.5098 |
hereis another shot at this after re thinking a bit
From wHolt - 11/21/06 1:04 PM
Why did you multiply by 4*14?
From TBird - 11/20/06 9:54 PM
EXPECTED PROBABILITY
| QUESTIONS | Answers | Why? | |
| 1 | How many ways can you deal a Blackjack? | 16 | Because AK AQ AJ A10 AK AQ AJ A10 AK AQ AJ A10 |
| 2 | How many Blackjack hands are there? | 16 | Because there are 16 ways you can deal a blackjack |
| 3 | How many ways can you deal 2 cards? | 112 | Because 14*4 = 56, 56*2 = 112 |
| 4 | How many 2 card hands are there? | 56 | Because 14*4 = 56 |
| 5 | What is the probability of dealing a Blackjack? (round off 3 places) | 14.29% | Because 16/112 = 0.142857 = 14.29% |
| 6 | You should expect a Blackjack about once in every WHOLE NUMBER hands. | 7 | Because 16/112 = 8/56 = 4/28 = 2/14 = 1/7 |
| 7 | How many Blackjacks should you expect in 104 hands? | 14 - 15? | 104 *.01429 = 14.8616? |
Last Modified 12/14/06 12:05 PM
#BJs
BJs/104
Obs-Exp
(Obs-Exp)/Exp
Warning! In my example, 6 is the wrong number.
You can figure out the right number by answering question 7,
or by observing class averages in column 8,
or by running the simulation a few times to see what you average.