Problem 79 - An Introduction to Mathematical Statistics and Its Applications

Question

A door-to-door encyclopedia salesperson is required to document five in-home visits each day. Suppose that she has a \(30 \%\) chance of being invited into any given home, with each address representing an independent trial. What is the probability that she requires fewer than eight houses to achieve her fifth success?

Step-by-Step Solution

Answer

The probability that the salesperson requires fewer than eight houses to achieve her fifth success is approximately calculated through steps 2 to 4 and is presented as the final result. Without actual calculation, the exact value cannot be given.

Step 1: Identify the parameters

Identify the parameters for the Negative Binomial Distribution. The number of successes, \(k\), we're interested in is 5 (being invited into 5 homes). The probability of success on each trial, \(p\), is \(30 \% = 0.3 \).

Step 2: Calculate cumulative probability

Calculate the cumulative probability for getting the fifth success on the first 7 houses using the Negative Binomial Distribution equation: \(P(X = k) = \binom{n-1}{k-1} \times p^k \times (1-p)^{n-k} \). Here, \(P(X = k)\) represents the probability of \(k\) success in \(n\) trials. The cumulative probability is the sum of the probabilities for getting the fifth success on the first, second, third, ..., or seventh house. As we want fewer than 8 houses, we do not include the eighth house in this calculation.

Step 3: Compute cumulative probability

Compute the cumulative probability. Starting from the fifth house up to the seventh house, calculate the cumulative probability and sum them. Apply the formula for each of the three cases.

Step 4: Final computation and rounding

Sum up the probabilities computed in Step 3. Round your final answer to an appropriate number of decimal places.

Key Concepts

Understanding Probability

Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, inclusive, where 0 indicates an impossibility, and 1 denotes certainty. In the context of the negative binomial distribution, the probability represents the chance of a certain number of successes occurring in a series of independent trials, before a specified number of failures. For instance, in our door-to-door encyclopedia salesperson example, the probability of success, being invited into a home, is given as 30% or as a decimal, 0.3.
To find the likelihood of the salesperson achieving exactly five successes, the negative binomial distribution considers both the probability of success in each trial and the number of trials required to achieve the fifth success. Calculating this requires us to determine the cumulative probability up to the point we are interested in, which in this case is before the salesperson reaches the eighth house.

Cumulative Probability in Practice

Cumulative probability refers to the probability that the variable takes on a value less than or equal to a certain point. In other words, it is the sum of probabilities of all outcomes up to and including that point. In our example, the salesperson needs five successes (invitations into homes), and we want to know the probability that she will achieve this after visiting fewer than eight houses. Therefore, we calculate the cumulative probability of getting the fifth success at the first, second, third, fourth, fifth, sixth, or seventh house.
To do this, we use the formula for the negative binomial distribution to find the probability for each required number of houses—five through seven—and then sum those probabilities. This gives us the cumulative probability of the salesperson achieving her fifth success before reaching the eighth house, and it is this sum that answers our exercise question.

Independent Trials and Their Significance

The concept of independent trials is essential to the negative binomial distribution and probability as a whole. An independent trial means that the outcome of one trial does not influence the outcomes of any other trials. For example, each home the salesperson visits is considered an independent trial since the outcome of whether she is invited in at one house does not affect her chances at the next house. The independence of trials ensures that the probability of success on each trial remains constant at 0.3.
Understanding that trials are independent allows us to apply the negative binomial distribution accurately because it assumes the consistency of probability across trials. In our context, ensuring that the condition of independence is met is essential for correctly modeling the scenario with the negative binomial distribution and calculating probabilities related to the number of trials needed to achieve a given number of successes.