Practice 2 Solutions

For problems 1–3, use the scenario.

A bag contains eight green marbles, five blue marbles, and seven red marbles. Determine the probability if marbles are drawn without replacement. Write your answer as a fraction.

$P (G, G, R)$

$\begin{array}{rcl} P (G, G, R) & = & \frac{8}{20} \cdot \frac{7}{19} \cdot \frac{7}{18} \\ = & \frac{49}{855} \end{array}$

$\frac{49}{855}$

$P (R, B, B)$

$\begin{array}{rcl} P (R, B, B) & = & \frac{7}{20} \cdot \frac{5}{19} \cdot \frac{4}{18} \\ = & \frac{7}{342} \end{array}$

$\frac{7}{342}$

$P (R, B, R)$

$\begin{array}{rcl} P (R, B, R) & = & \frac{7}{20} \cdot \frac{5}{19} \cdot \frac{6}{18} \\ = & \frac{7}{228} \end{array}$

$\frac{7}{228}$

For problems 4–6, use the scenario.

Your school is having a raffle, and there are 50 tickets sold. You (Y) and your friend (F) each buy three tickets. Only one ticket will be drawn for a $100 gift card, and a second ticket will be drawn for a $25 gift card. The first ticket drawn is not replaced. Determine the probability and write your answer to the tenth of a percent.

The probability that you win both prizes

$\begin{array}{rcl} P (Y, Y) & = & \frac{3}{50} \cdot \frac{2}{49} \\ = & \frac{3}{1225} \end{array}$

0.2%

The probability that you each win a prize

$\begin{array}{rcl} P (Y, F) + P (F, Y) \\ P (Y, F) + P (F, Y) & = & \frac{3}{50} \cdot \frac{3}{49} + \frac{3}{50} \cdot \frac{3}{49} \\ = & \frac{9}{1225} \end{array}$

0.7%

If you each bought 5 tickets, by how much would the probability of you both winning a prize increase?

$\begin{array}{rcl} P (Y, F) + P (F, Y) \\ P (Y, F) + P (F, Y) & = & \frac{5}{50} \cdot \frac{5}{49} + \frac{5}{50} \cdot \frac{5}{49} \\ = & \frac{1}{49} \\ = & 2.0 % \\ 2.0 % - 0.7 % & = & 1.3 % \end{array}$

The probability would increase by $1.3 % .$

For problems 7–9, use the scenario.

In a particular town, the probability of rain on any day is 25%. The probability of a traffic jam is 40%. The chance of a traffic jam given that it is raining is 70%.

What is the probability that it is raining and there is a traffic jam?

$\begin{array}{rcl} P (R and T) & = & P (R) \cdot P (T | R) \\ = & 0.25 \cdot 0.70 \\ = & 0.175 \end{array}$

17.5%

What is the probability of rain given a traffic jam?

$\begin{array}{rcl} P (R | T) & = & \frac{P (R \cap T)}{P (T)} \\ = & \frac{0.175}{0.40} \\ = & 0.4375 \end{array}$

43.75%

Note

The answer to problem 7 is needed in order to correctly solve problem 8.

Explain why the events are dependent.

$P (T and R) ? P (T) \cdot P (R) 0.70 \neq 0.40 \cdot 0.25$

The events are dependent because $P (T and R) \neq P (T) \cdot P (R)$ .

Note

To answer this problem, you need to compare the probability of a traffic jam with the conditional probability of a traffic jam given that it is raining. If the two probabilities are the same, the events are independent.

For problems 10–12, use the scenario.

Kristin wrapped up books on her summer reading list in identical wrapping paper. She is required to read three books. There are eight fiction (F) and five non-fiction (N) books. She randomly selects one wrapped book at a time to read. Determine the probability of selecting the books in the given order. Write the solution as a fraction.

$P (N, N, F)$

$\begin{array}{rcl} P (N, N, F) & = & \frac{5}{13} \cdot \frac{4}{12} \cdot \frac{8}{11} \\ = & \frac{40}{429} \end{array}$

$\begin{array}{rcl} \frac{40}{429} \end{array}$

Two fiction books and one non-fiction book

$\begin{array}{rcl} P (F, F, N) & = & \frac{8}{13} \cdot \frac{7}{12} \cdot \frac{5}{11} \\ = & \frac{70}{429} \end{array}$

$\begin{array}{rcl} \frac{70}{429} \end{array}$

$P (F, F, F)$

$\begin{array}{rcl} P (F, F, F) & = & \frac{8}{13} \cdot \frac{7}{12} \cdot \frac{6}{11} \\ = & \frac{28}{143} \end{array}$

$\begin{array}{rcl} \frac{28}{143} \end{array}$