Chapter 15, Problem 16E

(a)

To determine

To find: the probability for getting no aces.

Answer to Problem 16E

0.7826

Explanation of Solution

Given:

Total number of cards = 52

Ace cards = 4

Without Ace cards = 52-4=48

Calculation:

Required Probability

  $\begin{array}{c}P\left(\text{No}\text{aces}\right)=\frac{{}^{48}{C}_3}{{}^{52}{C}_3}\\ =\frac{\left(\frac{48!}{\left(48-3\right)!3!}\right)}{\left(\frac{52!}{\left(52-3\right)!3!}\right)}\left({}^n{C}_r=\frac{n!}{\left(n-r\right)!r!}\right)\\ =\left(\frac{48!}{45!3!}\right)\times \left(\frac{49!3!}{52!}\right)\\ =\left(\frac{48\times 47\times 46\times 45!}{45!}\right)\times \left(\frac{49!}{52\times 51\times 50\times 49!}\right)\left(n!=n\times \left(n-1\right)\times \left(n-2\right)\times \mathrm{...1}\right)\\ =\frac{48\times 47\times 46}{52\times 51\times 50}\\ =0.7826\end{array}$

Thus, the probability that for getting no aces in the three cards is 0.7826

(b)

To determine

To find: the probability for getting all hearts.

Answer to Problem 16E

0.0129

Explanation of Solution

Given:

Total number of cards = 52

Number of hearts = 13

Calculation:

Required probability

  $\begin{array}{c}P\left(\text{all three cards are hearts}\right)=\frac{13}{52}\times \frac{12}{51}\times \frac{11}{50}\left(\text{using multiplication rule}\right)\\ =\frac{1716}{132600}\\ =0.0129\end{array}$

Therefore, the probability that all three cards are hearts is 0.0129

(c)

To determine

To find: the probability that 3^rd card is first red card.

Answer to Problem 16E

0.1275

Explanation of Solution

Given:

Total number of cards = 52

Red cards =26

Calculation:

The required probability

  $\begin{array}{c}P\left(\text{Third cards is your first red card}\right)=\frac{26}{52}\times \frac{25}{51}\times \frac{26}{50}\left(\text{using multiplication rule}\right)\\ =\frac{16900}{132600}\\ =0.1275\end{array}$

Therefore, the probability that the third card is your first red card is 0.1275

(d)

To determine

To find: the probability that at least one diamond.

Answer to Problem 16E

0.5865

Explanation of Solution

Given:

Total number of Cards =52

Number of diamond cards =13

Calculation:

The required probability

  $\begin{array}{c}P\left(\text{at least one diamond}\right)=1-P\left(\text{no diamond}\right)\\ =1-\frac{39\times 38\times 37}{52\times 51\times 50}\\ =1-\frac{54834}{132600}\\ =1-0.4135\\ =0.5865\end{array}$

Therefore, the probability that you have at least one diamond is 0.5865