Chapter 15, Problem 15E

(a)

To determine

To find: the probability that the 1^st heart get is the 3^rd card dealt.

Answer to Problem 15E

0.1453

Explanation of Solution

Given:

Total number of cards = 52

Heart = 13

Other = 52-13= 39

Calculation:

Required to find the probability of getting first heart in the trial

  $\begin{array}{c}P\left(\begin{array}{l}\text{firs heart you get is}\\ \text{ the third card dealt}\end{array}\right)=P\left(\begin{array}{l}\text{first card is }\\ \text{not a heart}\end{array}\right)\times P\left(\begin{array}{l}\text{second card is }\\ \text{not a heart}\end{array}\right)\times P\left(\begin{array}{l}\text{Third card is }\\ \text{a heart}\end{array}\right)\\ =\frac{39}{52}\times \frac{38}{51}\times \frac{13}{50}\\ =\frac{19,266}{132,600}\\ =0.1453\end{array}$

(b)

To determine

To find: the probability that cards are all red.

Answer to Problem 15E

0.1176

Explanation of Solution

Given:

Total number of cards = 52

Red = 26

Calculation:

Required probability that all three drawn cards are red

  $\begin{array}{c}\left(\text{you cards are all red}\right)=\left[P\left(\text{first card is red}\right)\times P\left(\text{Second card is red}\right)\times P\left(\text{Third card is red}\right)\right]\\ =\frac{26}{52}\times \frac{25}{51}\times \frac{24}{50}\\ =\frac{15600}{132600}\\ =0.1176\end{array}$

The probability that all red is 0.1176

(c)

To determine

To find: the probability that get no spades.

Answer to Problem 15E

0.4135

Explanation of Solution

Given:

Total number of cards = 52

Spades = 13

Other = 52-13 =39

Formula used:

Calculation:

  $\begin{array}{c}P\left(\text{you get is no spades}\right)=\left[\begin{array}{l}P\left(\text{first card is not a spade}\right)\times \\ P\left(\text{Second card is not a spade}\right)\times \\ P\left(\text{Third card is not a spade}\right)\end{array}\right]\\ =\frac{39}{52}\times \frac{38}{51}\times \frac{37}{50}\\ =\frac{54834}{132600}\\ =0.4135\end{array}$

Probability that all cards dealt are not spade is 0.4135.

(d)

To determine

To find: the probability that at least one ace.

Answer to Problem 15E

0.2174

Explanation of Solution

Given:

Total number of cards = 52

Aces= 4

Calculation:

  $\begin{array}{c}P\left(\text{at least one ace}\right)=1-P\left(\text{No}\text{ace}\right)\\ =1-\left[\begin{array}{l}P\left(\text{first card is not an ace}\right)\times \\ P\left(\text{Second card is not an ace}\right)\times \\ P\left(\text{Third card is not an ace}\right)\end{array}\right]\\ =1-\left[\frac{48}{52}\times \frac{47}{51}\times \frac{46}{50}\right]\\ =1-\left[\frac{103776}{132600}\right]\\ =1-0.7826\\ =1-2174\end{array}$

The probability that at least one is Ace is 0.2174