Question 12.6E2: Using the Addition Formula Consider the same sample space, t......
Using the Addition Formula
Consider the same sample space, the numbers 1 through 10, as in Example 1. If one piece of paper is selected, determine the probability that it contains a number less than 5 or a number greater than 8.
Let A represent the statement “the number is less than 5” and B represent the statement “the number is greater than 8.” A Venn diagram illustrating these statements is shown in Figure 12.14.
\text{P(number is less than 5) = }\frac{4}{10}
\text{P(number is greater than 8) = }\frac{2}{10}
Since there are no numbers that are both less than 5 and greater than 8, P(number is less than 5 and greater than 8) = 0. Therefore,
\begin{aligned}& P \left( \begin{matrix} \text{number is}\\ \text{less than 5}\\ \text{or greater}\\ \text{than 8} \end{matrix} \right) = P \left( \begin{matrix} \text{number is}\\ \text{less than 5}\end{matrix} \right) + P \left( \begin{matrix} \text{number is}\\ \text{greater than 8} \end{matrix}\right) – P \left( \begin{matrix} \text{number is}\\ \text{less than 5}\\ \text{and greater}\\ \text{than 8} \end{matrix} \right)\end{aligned}
= \frac{4}{10} + \frac{2}{10} – 0 = \frac{6}{10} = \frac{3}{5}
Thus, the probability of selecting a number less than 5 or greater than 8 is \frac{3}{5}. The six numbers that are less than 5 or greater than 8 are 1, 2, 3, 4, 9, and 10.
