Chapter 5.3, Problem 101E

a.

To determine

To find: The probability that the first card dealt is a spade. Also, compute the conditional probability that the second card is a spade given first one is also spade.

Answer to Problem 101E

The probabilities are 0.25 and 0.2353 respectively.

Explanation of Solution

Given:

Number of spade cards in a deck = 13

Total number of cards = 52

Calculation:

The probability that the first card dealt is a spade is computed as:

  $\begin{array}{c}P\left(\text{Spade}\right)=\frac{\text{Number of spade cards}}{\text{Total number of cards}}\\ =\frac{13}{52}\\ =\frac{1}{4}\\ =0.25\end{array}$

The conditional probability that the second card is a spade given first one is also spade can be computed as:

  $\begin{array}{c}P\left(\begin{array}{l}\text{Second spade|}\\ \text{First spade}\end{array}\right)=\frac{P\left(\text{Second spade And First spade}\right)}{P\left(\text{First spade}\right)}\\ =\frac{\frac{13}{52}\times \frac{13}{51}}{\frac{13}{52}}\\ =0.2353\end{array}$

Thus, the required probabilities are 0.25 and 0.2353.

b.

To determine

To find: The conditional probabilities that second, third and fourth card are spade.

Answer to Problem 101E

The probabilities are 0.22, 0.20411 and 0.1875 respectively.

Explanation of Solution

Calculation:

The probabilities are:

  $\begin{array}{c}P\left(\begin{array}{l}\text{Second spade|}\\ \text{First spade}\end{array}\right)=\frac{P\left(\text{Second spade And First spade}\right)}{P\left(\text{First spade}\right)}\\ =\frac{\frac{13}{52}\times \frac{13}{51}}{\frac{13}{52}}\\ =0.2353\end{array}$

  $\begin{array}{c}P\left(\begin{array}{l}\text{Third spade|}\\ \text{Second spade}\end{array}\right)=\frac{P\left(\text{Third spade And Second spade}\right)}{P\left(\text{Second spade}\right)}\\ =\frac{\frac{13}{52}\times \frac{12}{51}\times \frac{11}{50}}{\frac{13}{52}\times \frac{12}{51}}\\ =0.22\end{array}$

  $\begin{array}{c}P\left(\begin{array}{l}\text{Forth spade|}\\ \text{Third spade}\end{array}\right)=\frac{P\left(\text{Forth spade And Third spade}\right)}{P\left(\text{Third spade}\right)}\\ =\frac{\frac{13}{52}\times \frac{12}{51}\times \frac{11}{50}\times \frac{10}{49}}{\frac{13}{52}\times \frac{12}{51}\times \frac{11}{50}}\\ =0.2041\end{array}$

  $\begin{array}{c}P\left(\begin{array}{l}\text{Fifth spade|}\\ \text{Forth spade}\end{array}\right)=\frac{P\left(\text{Forth spade And Fifth spade}\right)}{P\left(\text{Forth spade}\right)}\\ =\frac{\frac{13}{52}\times \frac{12}{51}\times \frac{11}{50}\times \frac{10}{49}\times \frac{9}{48}}{\frac{13}{52}\times \frac{12}{51}\times \frac{11}{50}\times \frac{10}{49}}\\ =0.1875\end{array}$

The probabilities are 0.22, 0.20411 and 0.1875 respectively.

c.

To determine

To find: The multiplication of all the probabilities.

Answer to Problem 101E

The value is 0.000495.

Explanation of Solution

Calculation:

The value can be calculated as:

  $\begin{array}{c}P\left(\text{Spade}\right)=\frac{1}{4}\times \frac{12}{51}\times \frac{11}{50}\times \frac{10}{49}\times \frac{9}{48}\\ =0.000495\end{array}$

The value is 0.000495.

d.

To determine

To find: The probability of having flush.

Answer to Problem 101E

The probability is 0.00198.

Explanation of Solution

Calculation:

The probability of having flush can be calculated as:

  $\begin{array}{c}P\left(\text{Flush}\right)=P\left(\text{5 spades}\right)+P\left(5\text{ Hearts}\right)+P\left(5\text{ Clubs}\right)+P\left(5\text{ Diamonds}\right)\\ =\frac{33}{66640}+\frac{33}{66640}+\frac{33}{66640}+\frac{33}{66640}\\ =\frac{132}{66640}\\ =0.00198\end{array}$

The probability is 0.00198.