Question

Raaj works at the customer service call center of a major credit card bank. Cardholders call for a variety of reasons, but regardless of their reason for calling, if they hold a platinum card, Raaj is instructed to offer them a double-miles promotion. About \(10 \%\) of all cardholders hold platinum cards, and about \(50 \%\) of those will take the double-miles promotion. On average, how many calls will Raaj have to take before finding the first cardholder to take the doublemiles promotion?

Short answer

Raaj will need to take, on average, 20 calls to find a cardholder who takes the double-miles promotion.

Step-by-step solution

Understanding the probability of a promotion acceptance

First, let's determine the probability of Raaj encountering a cardholder who holds a platinum card and accepts the double-miles promotion. The probability of randomly selecting a cardholder who holds a platinum card is given as \(0.10\) (10%). Once Raaj finds a platinum cardholder, the probability that this person will accept the promotion is \(0.50\) (50%). Therefore, the combined probability of both events happening (finding a platinum cardholder who accepts the promotion) is calculated by multiplying these probabilities together:\(P(\text{promotion}) = 0.10 \times 0.50 = 0.05\).

Applying the geometric distribution

Since we are interested in finding the first call in which a cardholder accepts the promotion, we use the geometric distribution. The geometric distribution models the number of trials needed to get the first success, where each trial is independent and has the same probability \(p\) of success.For the given problem, the probability of success on each call is \(p = 0.05\).

Calculating the expected number of calls

The expected number of trials (calls) needed to achieve the first success in a geometric distribution is given by the formula:\[ E(X) = \frac{1}{p} \]Where \(p\) is the probability of success on each trial.Substituting the value of \(p\) we found earlier:\[ E(X) = \frac{1}{0.05} = 20 \]

Conclusion: Result interpretation

This result means that, on average, Raaj will need to make 20 calls before finding the first platinum cardholder who opts for the double-miles promotion. This takes into account the probability combination of identifying a platinum cardholder and their acceptance of the offer.

Key concepts

Understanding Probability in Geometric Distribution

Probability plays a crucial role in determining the chances of an event happening. In the context of the geometric distribution, we're interested in the probability of "success" after several independent trials.
For the exercise, each call Raaj makes is considered a trial. Probability steps in to determine how likely Raaj is to encounter a platinum cardholder who also accepts the promotion. This is achieved by multiplying the probability of these two independent events:

• Probability of a cardholder having a platinum card: 0.10 or 10%.
• Probability of a platinum cardholder accepting the promotion: 0.50 or 50%.
Combining these, the probability (\( P_{ ext{promotion}} \)) that Raaj will find a platinum cardholder willing to accept the offer is \( 0.10 \times 0.50 = 0.05 \)or 5%.
This probability acts as the basis for the geometric distribution, which is key to calculating how many calls are expected till a promotion acceptance occurs.

Calculating Expected Value in Independent Trials

Expected value is a statistical measure used to determine the average number of trials needed to achieve a successful outcome. In a geometric distribution, this means identifying how many calls Raaj needs to make before someone accepts the promotion.
For independent trials—where each call is independent of others—the expected value (\( E(X) \)) gives us a clear insight into the average scenario. The formula for expected value in a geometric distribution is:

• \( E(X) = \frac{1}{p} \)
• \( p \)is the probability of success in each trial, here it is 0.05.

Substituting this probability in gives:\[ E(X) = \frac{1}{0.05} = 20 \]Thus, Raaj is expected to make 20 calls to achieve his first success, meaning the first platinum cardholder who takes the offer.

Role of Independent Trials in Geometric Distribution

Independent trials form the foundation of the geometric distribution. They assure that the outcome of any single trial does not affect the others. In Raaj's case, each call is an independent trial; no information from one call influences another. This independence is crucial because it keeps the promotional acceptance probability constant at 0.05. By considering the trials as independent, we can apply the geometric distribution model effectively. This means each call Raaj makes is a fresh, unaffected attempt to find a success. This is why it's valid to use the given probability formula and expect that, on average, it will take 20 calls to achieve the first success. Independent trials ensure a steady probability of success, providing a reliable framework for predicting outcomes in the context of a geometric distribution.