Question

Suppose that you are playing blackjack against a dealer. In a freshly shuffled deck, what is the probability that neither you nor the dealer is dealt a blackjack$?$

Short answer

the probability that neither you nor the dealer is dealt a blackjack is$0.90524$

Step-by-step solution

Step 1

let A be the player gets blackjack and B be the dealer gets a blackjack

P(A)$=$P(B)$=$$\left(4\times 16\right)/\left(\begin{array}{c}52\\ 2\end{array}\right)$

Probability that player and dealer gets blackjack is P(AB) role="math" localid="1647108119008" $\left[\left(4\times 16\right)/\left(\begin{array}{c}52\\ 2\end{array}\right)\right]\times \left[\left(3\times 15\right)/\left(\begin{array}{c}50\\ 2\end{array}\right)\right]$

Now, probability that neither you or dealer gets blackjack $=$$1-$P(A$\cup$B) $=1-$[P(A) $+$P(B)$-$P(AB)]$=1-\{2\times [(4\times 16)/\left(\begin{array}{c}52\\ 2\end{array}\right)]-[(64/\left(\begin{array}{c}52\\ 2\end{array}\right))\times (45/\left(\begin{array}{c}50\\ 2\end{array}\right))\left]\right\}=0.90524$