Chapter 11.3, Problem 44PPSE

Solution

a.

To identify: The probability that she pulls out a blue pen at randomTatyana has 2 blue pens. Two standard number cubes are rolled. What is the probability that the sum is greater than 9 or less than 62 Challenge 2x-1 pens in the pocket of his backpack.

The probability that $T$ pulls out a blue pen at random is $\frac{2}{x+2}$.

Given information:

The total number of pens with $T$ is 7 is $x+2$

The total number of pens with $S$ is $2x-1$ .

Explanation:

Consider the given information.

Objective is to find the probability that $T$ gets blue pen at random

Here, the total number of possible outcomes is $x+2$ , because $T$ has $x+2$ pens with her

The number of favorable outcomes is 2 because there are 2 blue pens

So, calculate the probability of getting blue pen at random as follows

  $\begin{array}{c}P\left(event\right)=\frac{Numberoffavorableoutcomes}{Numberofpossibleoutcomes}\\ P\left(gettingabluepen\right)=\frac{2}{x+2}\\ P\left(T\right)=\frac{2}{x+2}\end{array}$

Therefore, the probability that $T$ pulls out a blue pen at random is $\frac{2}{x+2}$.

b.

To identify:The probability that he pulls out a blue pen at random Samuel has x 3 blue pens.

The probability that $S$ pulls out a blue pen at random is $\frac{x-3}{2x-1}$

Given information:

The total number of pens with $T$ is 7 is $x+2$

The total number of pens with $S$ is $2x-1$ .

Explanation:

Consider the given information.

Objective is to find the probability that S gets a blue pen at random

Here, the total number of possible outcomes is $2x-1$ , because S has $2x-1$ pens with her

The number of favorable outcomes is $x-3$ , because there are $x-3$ blue pens

So calculate the probability of getting blue pen at random as follows

  $\begin{array}{c}P\left(event\right)=\frac{Numberoffavorableoutcomes}{Numberofpossibleoutcomes}\\ P\left(gettineabluepen\right)=\frac{x-3}{2x-1}\\ P\left(S\right)=\frac{x-3}{2x-1}\end{array}$

Therefore, the probability that $S$ pulls out a blue pen at random is $\frac{x-3}{2x-1}$

c.

To identify:The probability that either Tatyana or Samuel pulls out a blue pen at random.

  $270+1=271$ hexagonal cells.

Given information:

The total number of pens with $T$ is 7 is $x+2$

The total number of pens with $S$ is $2x-1$ .

Explanation:

Consider the given information.

Objective is to find the probability that either $TorS$ gets a blue pen at random.

Clearly, the events $T$ pulls out a blue pen and $S$ pulls out a blue pen are mutually exclusive.

So, the probability that either $T$ or $S$ gets a blue pen at random is sum of the probability that $T$ pulls out a blue pen at random and the probability that $S$ pulls out a blue pen at random.

This can be mathematically expressed as follows

  $\begin{array}{c}P\left(eitherTorSpullsoutbluepen\right)=P\left(T\right)+P\left(S\right)\\ =\frac{2}{x+2}+\frac{x-3}{2x-1}\\ =\frac{2\left(2x-1\right)+\left(x-3\right)\left(x+2\right)}{\left(x+2\right)\left(2x-1\right)}\\ =\frac{4x-2+x^2+2x-3x-6}{2x^2-x+4x-2}\\ =\frac{x^2+3x-8}{2x^2+3x-2}\end{array}$

Therefore, probability that either Tor S gets a blue pen at random is $\frac{x^2+3x-8}{2x^2+3x-2}$