Q. 2.18 - A First Course in Probability

Question

Two cards are randomly selected from an ordinary playing deck. What is the probability that it is a blackjack$?$ That is, what is the probability that one of the card is an ace and the other one is either a a ten, a jack, a queen or a king$?$

Step-by-Step Solution

Answer

The probability that it is a blackjack is $0.0483$

Step 1

Total number of ways to select two cards is $\left(\begin{array}{c}52\\ 2\end{array}\right)$.

Since, there are $4$ aces and $4\times 4=16$ tens, jacks, queens, kings the number of ways to select blackjack is $4\times 16=64$.

Thus, probability is $\raisebox{1ex}{$64$}\!\left/ \!\raisebox{-1ex}{$\left(\begin{array}{c}52\\ 2\end{array}\right)=0.0483$}\right.$

Step 2: Define the sample space and events

Identify the sample space and relevant events with their outcomes.

Step 3: Calculate individual probabilities

Compute the probability for each event using \(P(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}\).

Step 4: Apply probability rules

Use the appropriate rules (addition, multiplication, complement, or expected value formula \(E[X] = \sum x_i P(x_i)\)).

Step 5: State the final answer

Combine results to give the final probability or expected value.

Step 6: Conclude with the answer

The probability that it is a blackjack is $0.0483$