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In a game of blackjack/twenty-one between 2 players, the dealer and another player, the dealer shuffles a regular 52 card deck and deals himself 6H,6D and the player 8S,8C. What is the probability that the player does not go bust on the second hit (providing he didn't go bust on the first)? -Aces have a value of 1, face cards have a value of 10.

Dealer has 6h6d and player has 8s8c, so player's score before hitting is 16. To not bust on the first go, he needs a card lower than a 6. There are 2+4+2+4+16 = 28 cards that would make him go bust on the first go, out of 48 cards so probability of not busting first go = 20/48 = 5/12.

Chances for the second hit: All 28 cards that would have made him go bust on the first hit are still in the deck of now 47 cards. I have tried to go from here but all my methods end up with probabilities that cannot be correct as they do not have a sum total of 1. Any help would be appreciated, thanks.

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  • $\begingroup$ Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments. $\endgroup$ Sep 4 '18 at 9:32
  • $\begingroup$ @JoséCarlosSantos apologies, updated now (although not much help) $\endgroup$
    – user589837
    Sep 4 '18 at 9:43
  • $\begingroup$ Probability of going bust on the second hit after being dealt two $8$s and not splitting, if you never stick, is the probability of first hitting a $5$ followed by any card plus the probability of first hitting a $4$ followed by any card apart from an Ace plus the probability of first hitting a $3$ followed by any card apart from an Ace or $2$ plus the probability of first hitting a $2$ followed by any card apart from an Ace or $2$ or $3$ plus the probability of first hitting an Ace followed by any card apart from an Ace or $2$ or $3$ or $4$ $\endgroup$
    – Henry
    Sep 4 '18 at 10:02
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    $\begingroup$ Please do not vandalize your question by deleting the content. $\endgroup$
    – Xander Henderson
    Sep 13 '18 at 21:27
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    $\begingroup$ Please stop vandalizing your questions. Your behaviour is not appropriate, and is likely to earn you a suspension or ban. $\endgroup$
    – Xander Henderson
    Sep 13 '18 at 21:35
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The player doesn't go bust on the second hit if she draws a pair of cards that sum to $5$ or less. All cards up to $5$ are still in the deck. The admissible pairs are $(A,A)$, $(A,2)$, $(A,3)$, $(A,4)$, $(2,2)$, and $(2,3)$, which is $2$ pairs of equal ranks of which there are $\binom42=6$ each and $4$ pairs of different ranks of which there are $4^2=16$ each, for a total of $2\cdot6+4\cdot16=76$ pairs out of $\binom{48}2=1128$, so the probability is $\frac{76}{1128}=\frac{19}{282}\approx6.7\%$.

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