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Chapter 5: Problem 40

A driving exam consists of 30 multiplechoice questions. Each of the answers is either right or wrong. Suppose that the probability of making fewer than 7 mistakes is \(0.23\) and the probability of making from 7 to 15 mistakes is \(0.41\). Find the probability of making the following: a. 16 or more mistakes b. 7 or more mistakes c. At most 15 mistakes d. Which two of these three events are complementary? Explain.

Short Answer

Expert verified
a. Probability of making 16 or more mistakes is \(0.36\)\n b. Probability of making 7 or more mistakes is \(0.77\)\n c. Probability of making at most 15 mistakes is \(0.64\)\n d. The pair 'making 7 or more mistakes' and 'making at most 15 mistakes' are complementary events.

Step by step solution

01

Calculate Probability of Making 16 or More Mistakes

The probability of making fewer than 7 mistakes is \(0.23\) and the probability of making from 7 to 15 mistakes is \(0.41\). The total is therefore \(0.23 + 0.41 = 0.64\). Since the sum of the probabilities of all possibilities is always 1, to find the probability of making 16 or more mistakes, subtract this value from 1: \(1 - 0.64 = 0.36\). So, the probability of making 16 or more mistakes is \(0.36\).
02

Calculate Probability of Making 7 or More Mistakes

To find the probability of making 7 or more mistakes, simply add the probability of making from 7 to 15 mistakes and the probability of making 16 or more mistakes that was calculated in step 1: \(0.41 + 0.36 = 0.77\). So, the probability of making 7 or more mistakes is \(0.77\).
03

Calculate Probability of Making At Most 15 Mistakes

To find the probability of making at most 15 mistakes, add the probability of making fewer than 7 mistakes and the probability of making from 7 to 15 mistakes: \(0.23 + 0.41 = 0.64\). So, the probability of making at most 15 mistakes is \(0.64\).
04

Determine Complementary Events

Two events are complementary if one event happening prohibits the other. Meaning, the sum of their probabilities is 1. From the calculations above, the pair of results with the sum equal to 1 is 'making 7 or more mistakes' and 'making at most 15 mistakes'. Therefore, these two events are complementary.

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