Chapter 13.5, Problem 11CFU

To determine

To calculate: The probability that numbers match or sum is even when a pair of number cubes is thrown provided that their sum is greater than or equal to 9.

Answer to Problem 11CFU

The probability that numbers match or sum is even when a pair of number cubes is thrown provided that their sum is greater than or equal to 9is $\frac{2}{5}$ .

Explanation of Solution

Given information:

A pair of number cubes is thrown.

Formula used:

Probability of an event Eis $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Given an event Q , the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

Calculation:

Consider the provided information thata pair of number cubes is thrown.

Sample space of the event is $S=\left\{\begin{array}{cccccc}\left(1,1\right)& \left(1,2\right)& \left(1,3\right)& \left(1,4\right)& \left(1,5\right)& \left(1,6\right)\\ \left(2,1\right)& \left(2,2\right)& \left(2,3\right)& \left(2,4\right)& \left(2,5\right)& \left(2,6\right)\\ \left(3,1\right)& \left(2,3\right)& \left(3,3\right)& \left(3,4\right)& \left(3,5\right)& \left(3,6\right)\\ \left(4,1\right)& \left(2,4\right)& \left(4,3\right)& \left(4,4\right)& \left(4,5\right)& \left(4,6\right)\\ \left(5,1\right)& \left(2,5\right)& \left(5,3\right)& \left(5,4\right)& \left(5,5\right)& \left(5,6\right)\\ \left(6,1\right)& \left(2,6\right)& \left(6,3\right)& \left(6,4\right)& \left(6,5\right)& \left(6,6\right)\end{array}\right\}$

The probability that numbers match or sum is even when a pair of number cubes is thrown provided that their sum is greater than or equal to 9.

Let A be an event that sum is even and B be an event that sum is greater than or equal to 9.

Recall that probability of an event Eis $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Probability that sum is greater than or equal to 9,

  $P\left(B\right)=\frac{10}{36}$

Probability that numbers match or sum is even and sum is greater than or equal to 9,

  $P\left(A\text{ and B}\right)=\frac{4}{36}$

Recall that given an event Q , the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

The probability that sum is even when a pair of number cubes is thrown provided that their sum is greater than or equal to 9.

  $\begin{array}{c}P\left(A|B\right)=\frac{\frac{4}{36}}{\frac{10}{36}}\\ =\frac{4}{10}\\ =\frac{2}{5}\end{array}$

Thus, the probability that numbers match or sum is even when a pair of number cubes is thrown provided that their sum is greater than or equal to 9is $\frac{2}{5}$ .