To determine the probability that two valuable cards (Jack, Queen, King, or 10) are dealt consecutively in a game of blackjack, we need to follow a step-by-step approach. Here is a detailed solution to this question:
1. Identify the Number of Valuable Cards: In a standard deck of 52 cards, the valuable cards include Jacks, Queens, Kings, and 10s. Each of these four types appears in each of the four suits (hearts, diamonds, clubs, spades). Therefore, the total number of valuable cards in the deck is:
[tex]\[
\text{Number of valuable cards} = 4 \text{ (Jacks)} + 4 \text{ (Queens)} + 4 \text{ (Kings)} + 4 \text{ (10s)} = 16
\][/tex]
2. Calculate the Probability of Drawing the First Valuable Card: The total number of cards in the deck is 52. When drawing the first card, the probability that it is one of the 16 valuable cards is:
[tex]\[
P(\text{First card is valuable}) = \frac{16}{52} \approx 0.307692
\][/tex]
3. Calculate the Probability of Drawing the Second Valuable Card, Given the First Was Valuable: After drawing the first valuable card, we have one less valuable card and one less total card in the deck. Therefore, there are now 15 valuable cards remaining out of the remaining 51 cards. The probability of drawing a valuable card as the second card, given that the first card was valuable, is:
[tex]\[
P(\text{Second card is valuable} \mid \text{First card is valuable}) = \frac{15}{51} \approx 0.294118
\][/tex]
4. Calculate the Joint Probability of Both Cards Being Valuable: To find the joint probability that both the first and second cards are valuable, we multiply the individual probabilities:
[tex]\[
P(\text{Both cards are valuable}) = P(\text{First card is valuable}) \times P(\text{Second card is valuable} \mid \text{First card is valuable})
\][/tex]
[tex]\[
P(\text{Both cards are valuable}) \approx 0.307692 \times 0.294118 \approx 0.090498
\][/tex]
Therefore, the probability that two valuable cards are dealt consecutively in a game of blackjack is approximately 0.090498, or 9.05%.
