NUMB3RS: "Double Down" — Blackjack & Dependent Events

Blackjack is essentially a card game between a player and a dealer. To win, a player’s hand must have a value closer to $21$ than the dealer’s hand, without going over $21$. Aces are worth either $1$ or $11$, face cards ($J, Q, K$) and tens are worth $10$, and all other cards are worth their face value.

Because cards are drawn from the deck and not replaced, blackjack probabilities are based on sampling without replacement and dependent trials.

For example, to find the probability of being dealt two Aces in a row:

1. The probability of the first card being an Ace is $P(\text{Ace}_1) = \frac{4}{52}$.
2. Since the card is not replaced, there are now $51$ cards left, containing $3$ Aces. The probability of the second card being an Ace is $P(\text{Ace}_2 \mid \text{Ace}_1) = \frac{3}{51}$.
3. Multiplying these gives: $P(\text{Ace and Ace}) = \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} \approx 0.0045$