Q.7.18 - A First Course in Probability

Question

Cards from an ordinary deck of $52$ playing cards are turned face upon at a time. If the 1st card is an ace, or the $2$nd a deuce, or the $3$rd a three, or ...,or the $13$th a king,or the $14$ an ace, and so on, we say that a match occurs. Note that we do not require that the ($13$n + $1$) card be any particular ace for a match to occur but only that it be an ace. Compute the expected number of matches that occur.

Step-by-Step Solution

Answer

The expected number of matches that occur is 4

$E\left(N\right)=4$

Step 1: Given Information

Let Xi be the event that when we turn over card i if matches the required cards face.

For example, X$1$ is the event that turning over card one reveals an ace.

X$2$ is the event that turning over the second card reveals a deuce etc.

Step 2: Explanation

The number of matched cards N is given by the sum of these indicator random variables as,

$N=\sum _{i=1}^{52}{X}_i$

$P\left(X_i\right)=\frac{4}{52}\text{, probability that a certain selection is a match is,}$

$P\left(X_i\right)=\frac{4}{52}=\frac{1}{13}$

$E\left(N\right)=\sum _{i=1}^{52}E\left(X_i\right)$

$=\sum _{i=1}^{52}P\left(X_i\right)$

$=52.\frac{1}{13}$

$=4$

Step 3: Final Answer

The expected number of matches that occur is $4$.

$E\left(N\right)=4$