A factory makes computer hard disks. Over a long period, 1.5...

A factory makes computer hard disks. Over a long period, $1.5 \%$ of them are found to be defective. A random sample of 50 hard disks is tested. a) Write down the expected number of defective hard disks in the sample. b) Find the probability that three hard disks are defective. c) Find the probability that more than one hard disk is defective.

A factory makes computer hard disks. Over a long period, $1.5 \%$ of them are found to be defective. A random sample of 50 hard disks is tested.
a) Write down the expected number of defective hard disks in the sample.
b) Find the probability that three hard disks are defective.
c) Find the probability that more than one hard disk is defective.

Key Concepts

Binomial Distribution

The binomial distribution is used to model the number of successes in a fixed number of independent trials where each trial has the same probability of success. In this problem, each hard disk is an independent trial with two possible outcomes—defective or not defective—making the binomial distribution an appropriate model for determining probabilities related to defectiveness.

Expected Value

The expected value in the context of a binomial distribution is the product of the number of trials and the probability of success. It provides a measure of the average outcome one would expect over many trials, which in this case gives the average number of defective hard disks in a sample.

Probability Mass Function

The probability mass function (PMF) of a binomial distribution gives the probability of obtaining exactly a specified number of successes. It is calculated using the formula P(X = k) = (n choose k) * p^k * (1 - p)^(n - k), where 'n' is the total number of trials, 'p' is the probability of success, and 'k' is the number of successes.

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability of it not occurring. This rule can simplify the process of calculating the probability of events described in terms like 'more than one' by subtracting the probability of the complementary events from one.

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