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
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
