Q.7.35 - A First Course in Probability

Question

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.

Step-by-Step Solution

Answer

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

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

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

Step 1: Given Information (Part a)

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 $=?$

Step 2: Explanation (Part a)

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 $i^{\text{th }}$Ace card is,

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

Step 3: Explanation (Part a)

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

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

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

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

$=\frac{106}{5}$

$=21.2$

Step 4: Final Answer (Part a)

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

Step 1: Given Information (Part b)

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 =?

Step 2: Explanation (Part b)

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, 

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

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

$=18.92857$

$=18.929$

Step 3: Final Answer (Part b)

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

Step 1: Given Information (Part c)

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=?

Step 2: Explanation (Part c)

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,

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

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

$=\frac{53}{14}$

Step 3: Explanation (Part c)

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

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

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

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

$=49.21429$

$=49.214$

Step 4: Final Answer (Part c)

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