Probability of being delt a BJ

B

BlueLight

Active Member
In a 1 deck game the cards are delt face down. You get 2 cards face down and the dealer gets 2 cards face down before turning his first delt card face up.

1) Before the dealer turns his first delt card face up; What is the probability you have been delt a blackjack?

2) After he turns his first delt card face up and reveals an "8"; What is the probability you have been delt a blackjack?

3) After the dealer turns up his 1st delt card and reveals an "A" and before he peeks to check for a dealer blackjack; What is the probability you have been delt a blackjack.

4) After the dealer turns up his 1st delt card and reveals an "A" and after he peeks to check for a BJ and doesn't turn over a BJ; What is the probability you have been delt a BJ?

Get the answers as an exact fraction or as a decimal with 6 figures after the decimal point?
Parts 1-3 are relatively easy but part 4 is a little tricky.
This teaser is a little more difficult than the "PEEK EFFECT" teaser I posted long ago.


...........................................................BlueLight
 
Hint: Assume the player doesn't get his cards until after the dealer has delt his (the dealer's) cards and has checked for a BJ if necessary.

For example in situation 2) above an 8 is missing from the deck, leaving a 51 card deck with 3 8's. Now what is the probability the next 2 cards delt (to the player) will be a BJ?

The player could be delt an A and then a 10; for a probability of 4/51 x 16/50 = 64/2550 = 32/1275

And the player could be delt a 10 and then an A; for a probability of 16/51 x 4/50 = 64/2550 = 32/1275

Combining these 2 ways gets 64/1275 = .050196+


Another way would be to cycle through different dealer down cards:

The dealer has an A hole card with a probability of 4/51
This gets hole card = A and a A+10 BJ for 4/51 x 3/50 x 16/49 = 192 /124,950
Also gets hole card = A and a 10+A BJ for 4/51 x 16/50 x 3/49 = 192/124,950

The dealer has a 10 hole card with a probability of 16/51
This gets hole card = 10 and a A+10 BJ for 16/51 x 4/50 x 15/49 = 960/124,950
Also gets hole card = 10 and a 10+A BJ for 16/51 x 15/50 x 4/49 = 960/124,950

The dealer has a hole card of 2 thru 9 with a probability of 31/51 (remember only 3 8's left)
This gets hole card = 2-9 and a A+10 BJ for 31/51 x 4/50 x 16/49 = 1984/124,950
Also gets hole card = 2-9 and a 10+A BJ for 31/51 x 16/50 x 4/49 = 1984/124,950

Adding all these up gets 6272/124,950. Divide numerator and denominator by 49 gets 64/1275

To solve the problem 4) you need to cycle thru "possible" dealer hole cards. The hole card cannot be a 10. The possible dealer hole cards can be of 35 possible cards.


Hope this makes the puzzle easier.


..........................................................BlueLight
 
Last edited:
To figure part 4 for the player probability of having a BJ when the dealer up card is an A the following table is usefull.

Card type...................................A......2......3......4......5......6......7......8......9......10
Unseen cards.............................3......4......4......4......4......4......4......4......4......16 = 51
Possible dealer down Card........3......4......4......4......4......4......4......4......4.......0 = 35

Now the probability the dealer has an A down card is 3/35
With an A down card the prob of player getting an A and then a 10 = 2/50 x 16/49
With an A down card the prob of player getting a 10 and then an A = 16/50 x 2/49

Therefore the total probability of the player getting a BJ and dealer having A down card is:

3/35 x (2) x 2/50 x 16/49 = 192/85750 = 96/42875

Now the probability the dealer has a 2-9 down card is 32/35
With a 2-9 down card the prob of player getting an A and then a 10 = 3/50 x 16/49
With a 2-9 down card the prob of player getting a 10 and then an A = 16/50 x 3/49

Therefore the total probability of the player getting BJ and dealer having a 2-9 down card is:

32/35 x (2) x 3/50 x 16/49 = 3072/85750 = 1536/42875

There is no need to calculate the player probability of getting a BJ with a dealer down card of a 10 since that cannot occur after checking for a BJ and he does not have the BJ.

Adding the 2 fractions to get the total player probability of getting a BJ against all possible down cards gets:

(96 + 1536)/42875 = 1632/42875 = .0380641399+


...........................................BlueLight
 
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