Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards that need to be turned face up in order to obtain

(a) 2 aces;

(b) 5 spades;

(c) all 13 hearts.

Cards from an ordinary deck are turned face up one at a time.

The expected number of cards that need to be turned face up in order to obtain 2 aces$=?$

Let $X_i$be the number that need to be turned over before an $i^{\text{th}}$Ace card appears.

Let $n_i$be the number of cards of type $i$of total of$n$cards.

Here, the number of aces in a deck of cards is,

$m=4$

The total number of cards in deck of cards is,

$n=52$

Now, the expected value of $i^{\text{th}}$Acecard is,

$E\left(X_i\right)=\frac{n+1}{m+1}$

The expected number of cards that need to be turned face up in order to obtain 2 aces is,

$E\left(X\right)=2E\left(X_i\right)$

$=2\times \frac{52+1}{4+1}$

$=2\left(\frac{53}{5}\right)$

$=\frac{106}{5}$

$=21.2$

The expected number of cards that need to be turned face up in order to obtain 2 aces is, $21.2$

Cards from an ordinary deck are turned face up one at a time.

The expected number of cards that need to be turned face up in order to obtain 5 spades is =?

Let $X_i$be the number that need to turn over before $a^{\text{th}}$spade appears.

So, the expected value of $i^{\text{th}}$spade appears is,

$E\left(X_i\right)=\frac{52+1}{13+1}$

$=\frac{53}{14}$

The expected number of cards that need to be turned face up in order to obtain 5 spades is,

$\begin{array}{r}E\left(X\right)=5E\left(X_i\right)\end{array}$

$\begin{array}{r}=5\left(\frac{53}{14}\right)\end{array}$

role="math" localid="1647520376889" $=18.92857$

$=18.929$

The expected number of cards that need to be turned face up in order to obtain 5 spades is$18.929$

Cards from an ordinary deck are turned face up one at a time.

The expected number of cards that need to be turned face up in order to obtain 13 hearts=?

Let $X_i$be the number of cards that need to turn over before $i^{\text{th}}$Heart appears.

Let $X$be the number of cards that need be turned face up in order to obtain all 13 Hearts.

So,

$X=X_1+X_2+\dots +X_{13}$

Let the number of spade cards be $m=13$

The expected number of cards that need to be turned face up in order to obtain $i^{th}$heart is,

$\begin{array}{r}E\left(X_i\right)=\frac{n+1}{m+1}\end{array}$

role="math" localid="1647520571977" $=\frac{52+1}{13+1}$

$=\frac{53}{14}$

The expected number of cards that need to be turned face up in order to obtain 13 hearts is,

$\begin{array}{r}E\left(X\right)=13E\left(X_i\right)\end{array}$

$\begin{array}{r}=13\times \frac{53}{14}\end{array}$

$\begin{array}{r}=\frac{689}{14}\end{array}$

localid="1647520773126" $=49.21429$

$=49.214$

The expected number of cards that need to be turned face up in order to obtain 13 hearts is$49.214$