EXAMPLE 5.8 Determining Consumer Preferences

Suppose that market research shows that your product is preferred over competitors’ products by 25% of all consumers. If is the count of the number of consumers who prefer your product in a group of five consumers, then has a binomial distribution with and , provided the five consumers make choices independently. What is the probability that exactly two consumers in the group prefer your product? We are seeking .

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Because the method doesn’t depend on the specific example, we will use “S” for success and “F” for failure. Here, “S” would stand for a consumer preferring your product over the competitors’ products. We do the work in two steps.

Step 1. Find the probability that a specific two of the five consumers—say, the first and the third—give successes. This is the outcome SFSFF. Because consumers are independent, the multiplication rule for independent events applies. The probability we want is

Step 2. Observe that the probability of any one arrangement of two S’s and three F’s has this same probability. This is true because we multiply together 0.25 twice and 0.75 three times whenever we have two S’s and three F’s. The probability that is the probability of getting two S’s and three F’s in any arrangement whatsoever. Here are all the possible arrangements:

There are 10 of them, all with the same probability. The overall probability of two successes is therefore

Approximately 26% of the time, samples of five independent consumers will produce exactly two who prefer your product over competitors’ products.