Chapter 13.5, Problem 4CFU

To determine

To calculate: The probability that numbers match when a pair of number cubes is thrown provided that their sum is greater than 5.

Answer to Problem 4CFU

The probability that numbers match when a pair of number cubes is thrown provided that their sum is greater than 5is $\frac{2}{13}$ .

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 when a pair of number cubes is thrown provided that their sum is greater than 5.

Let A be an event that numbers match 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 5,

  $P\left(B\right)=\frac{26}{36}$

Probability that numbers match and sum is greater than 5,

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

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 numbers match when a pair of number cubes is thrown provided that their sum is greater than 5.

  $\begin{array}{c}P\left(A|B\right)=\frac{\frac{4}{36}}{\frac{26}{36}}\\ =\frac{4}{26}\\ =\frac{2}{13}\end{array}$

Thus, the probability that numbers match when a pair of number cubes is thrown provided that their sum is greater than 5 is $\frac{2}{13}$ .