Problem 34 - Algebra and Trigonometry

Question

Telephone Marketing A mortgage company advertises its rates by making unsolicited telephone calls to random numbers. About 2\(\%\) of the calls reach consumers who are interested in the company's services. A telephone consultant can make 100 calls per evening shift. (a) What is the probability that two or more calls will reach an interested party in one shift? (b) How many calls does a consultant need to make to ensure at least a 0.5 probability of reaching one or more interested parties? IHint: Use trial and error.

Step-by-Step Solution

Answer

(a) Probability of two or more interested calls is about 0.677. (b) Approximately 35 calls are needed for at least a 0.5 probability.

Step 1: Identify the Probability Distribution

This problem involves making random calls, which follows a binomial distribution where each call is an independent trial. The probability of success (a consumer being interested) is \( p = 0.02 \).

Step 2: Calculate Probability of Two or More Successes

First, calculate the probability of zero or one success in 100 calls. This will help us find the probability of two or more successes using the complement rule.The binomial probability is given by the formula \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \).For zero successes:\[P(X = 0) = \binom{100}{0} (0.02)^0 (0.98)^{100} = (0.98)^{100}\]For one success:\[P(X = 1) = \binom{100}{1} (0.02)^1 (0.98)^{99} = 100 \times 0.02 \times (0.98)^{99}\]Finally, sum these probabilities to find \( P(X < 2) \):\[P(X < 2) = P(X = 0) + P(X = 1)\]The probability of having two or more interested consumers is:\[P(X \geq 2) = 1 - P(X < 2)\]

Step 3: Calculate the Required Number of Calls for 0.5 Probability

To ensure at least a 0.5 probability of reaching one or more interested parties, we use trial and error with the formula for \( P(X \geq 1) = 1 - P(X = 0) \).Starting with a smaller number of calls (n), solve:\[P(X \geq 1) = 1 - (1 - 0.02)^n \geq 0.5\]Testing different values of \( n \), starting from 10 onwards, find the smallest number for which \( 1 - (0.98)^n \geq 0.5 \).

Step 4: Compute Results for (a) and (b)

(a) After calculating \( P(X < 2) \) using \((0.98)^{100}\) and \(100 \times 0.02 \times (0.98)^{99}\), you get approximately \( P(X < 2) \approx 0.323 \), hence \( P(X \geq 2) \approx 0.677 \).(b) For starting with \( n = 35 \), check that \( P(X \geq 1) \approx 1 - (0.98)^{35} \approx 0.502 \), suggesting that about 35 calls are needed to achieve the 0.5 probability.

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the likelihood of events happening. It is based on the idea that we can assign a number to how probable an event is, ranging from 0 (impossible event) to 1 (certain event). In this exercise, we explore how probability theory applies to real-world scenarios such as telephone marketing calls.
Advertisers, like the mortgage company in the problem, often use probability to predict the outcomes of their campaigns. When making a call, there are two possible outcomes: the consumer is either interested or not interested. Here, the probability of a successful call (where the consumer is interested) is 2% or 0.02.
In probability theory, these types of experiments are called trials, and each call represents a trial which is independent of others. That means the outcome of one call does not affect the outcome of another. Probability theory provides tools to compute outcomes across many trials, helping companies to predict how many interested consumers they might reach during a call campaign.

Statistical Analysis

Statistical analysis involves collecting, reviewing, interpreting, and drawing conclusions from data. In this problem, statistical analysis is used to determine the likelihood of different outcomes when making telephone calls. Using statistical methods, we can calculate the probabilities of various numbers of interested consumers reached within a given number of calls.
A key component here is the binomial distribution, which is ideal for analyzing scenarios with two outcomes, such as success (an interested consumer) or failure (an uninterested consumer). We use statistical formulas to calculate probabilities associated with different numbers of successes, like getting two or more interested calls.

• For zero successes in 100 calls, we calculate the probability using: \[P(X = 0) = (0.98)^{100}\]
• Similarly, for one success, the formula is:\[P(X = 1) = 100 \times 0.02 \times (0.98)^{99}\]

Through statistical analysis, we then sum these results to discover that the probability of two or more interested parties exceeds 67%. This demonstrates how statistical techniques can give insights into business strategies and expected outcomes.

Mathematical Modeling

Mathematical modeling is the process of representing real-world problems with mathematical structures and relationships. It is a powerful tool used to simulate situations, estimate outcomes, and make informed decisions.
In the context of the given exercise, mathematical modeling is leveraged to understand the calling strategy's effectiveness. By using a binomial model, we represent the random variable, number of interested calls, and set up equations to solve the problem conditions using the parameters given.
We model the scenario of needing a 0.5 probability of reaching at least one interested party through trial and error by adjusting the number of calls. The critical modeling question is to find out how many calls ensure a probability of success at or above 50%.

• We start solving for:\[P(X \geq 1) = 1 - (0.98)^n \geq 0.5\]
• By testing different values of \(n\), it is deduced that approximately 35 calls yield the required probability.

This mathematical interpretation helps in establishing sound decision-making processes in marketing, allowing advertisers to adjust strategies based on predicted outcomes.