14.5 Examples of the Negative Binomial Distribution

The Negative Binomial Distribution is a discrete probability distribution that models the probability of achieving a specific number of successes in a sequence of independent trials, each with the same probability of success. It is particularly useful when you're interested in the number of trials required to achieve a certain number of successes.

Formula

The probability mass function (PMF) of the Negative Binomial distribution (where $x$ is the total number of trials and $k$ is the number of successes) is given by:

$P(X = x) = \binom{x - 1}{k - 1} \theta^k (1 - \theta)^{x - k}$

Where:

• $P(X = x)$: The probability of achieving the $k$-th success on the $x$-th trial.
• $x$: The total number of trials (where $x \ge k$).
• $k$: The number of successes needed.
• $\theta$: The probability of success on a single trial.
• $1 - \theta$: The probability of failure on a single trial.
• $\binom{x - 1}{k - 1}$: The binomial coefficient, representing the number of ways to arrange the successes and failures such that the $k$-th success occurs on the $x$-th trial. It is calculated as $\frac{(x-1)!}{(k-1)!(x-k)!}$.

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Solved Examples

Here are two solved examples illustrating the application of the Negative Binomial Distribution.

Example 1: Puzzle Solving Probability

Problem:

John has an 80% probability of solving a puzzle correctly on any given attempt. What is the probability that John solves the puzzle correctly for the fourth time on his seventh attempt?

Solution:

1. Identify the parameters:

   • Number of successes needed ($k$): 4
   • Total trials ($x$): 7
   • Probability of success ($\theta$): 80% or 0.80
   • Probability of failure ($1 - \theta$): 1 - 0.80 = 0.20

2. Apply the Negative Binomial Probability Formula:

   $P(X = x) = \binom{x - 1}{k - 1} \theta^k (1 - \theta)^{x - k}$

3. Substitute the values:

   $P(X = 7) = \binom{7 - 1}{4 - 1} (0.80)^4 (0.20)^{7 - 4}$ $P(X = 7) = \binom{6}{3} (0.80)^4 (0.20)^3$

4. Calculate the components:

   • Binomial Coefficient: $\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$
   • Probability of Successes: $(0.80)^4 = 0.4096$
   • Probability of Failures: $(0.20)^3 = 0.008$

5. Compute the final probability:

   $P(X = 7) = 20 \times 0.4096 \times 0.008$ $P(X = 7) = 0.065536$

Answer: The probability that John solves the puzzle correctly for the fourth time on his seventh attempt is approximately 0.066 or 6.6%.

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Example 2: Archery Success Probability

Problem:

Emma has an 85% probability of hitting the target in archery on any given shot. What is the probability that Emma hits the target for the third time on her fifth shot?

Solution:

1. Identify the parameters:

   • Number of successes needed ($k$): 3
   • Total trials ($x$): 5
   • Probability of success ($\theta$): 85% or 0.85
   • Probability of failure ($1 - \theta$): 1 - 0.85 = 0.15

2. Apply the Negative Binomial Probability Formula:

   $P(X = x) = \binom{x - 1}{k - 1} \theta^k (1 - \theta)^{x - k}$

3. Substitute the values:

   $P(X = 5) = \binom{5 - 1}{3 - 1} (0.85)^3 (0.15)^{5 - 3}$ $P(X = 5) = \binom{4}{2} (0.85)^3 (0.15)^2$

4. Calculate the components:

   • Binomial Coefficient: $\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6$
   • Probability of Successes: $(0.85)^3 \approx 0.614125$
   • Probability of Failures: $(0.15)^2 = 0.0225$

5. Compute the final probability:

   $P(X = 5) = 6 \times 0.614125 \times 0.0225$ $P(X = 5) \approx 0.08244375$

Answer: The probability that Emma hits the target for the third time on her fifth shot is approximately 0.082 or 8.2%.

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Conclusion

The Negative Binomial Distribution is a versatile tool for modeling scenarios involving repeated trials where the focus is on the number of trials needed to achieve a specific count of successes. It finds applications in various fields, including:

• Games and Sports: Analyzing the number of attempts to win a certain number of matches.
• Quality Control: Determining how many items to inspect before finding a specified number of defective ones.
• Marketing: Estimating the number of customer contacts required to secure a certain number of sales.
• Project Management: Assessing the number of tasks to complete to reach a milestone.

The core of its application lies in understanding the probability of the k-th success occurring on the x-th trial, given a constant probability of success for each independent trial.

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Interview Questions

Here are some common interview questions related to the Negative Binomial Distribution:

1. What is the negative binomial distribution and when is it used?

   • Answer: It's a discrete probability distribution that describes the probability of the number of trials required to achieve a specified number of successes in a series of independent Bernoulli trials, each with the same probability of success. It's used when the number of trials is not fixed, but the number of successes is.

2. Can you state the formula for the negative binomial probability?

   • Answer: $P(X = x) = \binom{x - 1}{k - 1} \theta^k (1 - \theta)^{x - k}$, where $x$ is the number of trials, $k$ is the number of successes, and $\theta$ is the probability of success.

3. How do you interpret the parameters $k$ and $x$ in the negative binomial distribution?

   • Interpretation: $k$ represents the target number of successes you aim to achieve. $x$ represents the total number of trials conducted, with the condition that the $k$-th success occurs precisely on the $x$-th trial.

4. Explain the role of the binomial coefficient in the negative binomial formula.

   • Explanation: The binomial coefficient $\binom{x - 1}{k - 1}$ accounts for all the possible sequences of successes and failures that can occur in the first $x-1$ trials, leading up to the $k$-th success on the $x$-th trial. It counts the number of ways to arrange $k-1$ successes (and consequently $x-k$ failures) within the first $x-1$ trials.

5. How do you calculate the probability of achieving the $k$-th success on the $x$-th trial?

   • Calculation: You use the formula $P(X = x) = \binom{x - 1}{k - 1} \theta^k (1 - \theta)^{x - k}$, by plugging in the values for $x$, $k$, and $\theta$.

6. Can you solve an example problem involving the negative binomial distribution?

   • (This question tests practical application, similar to the examples provided above.)

7. What assumptions are made about trials in the negative binomial distribution?

   • Assumptions:
     • Each trial is independent of the others.
     • Each trial has only two possible outcomes: success or failure.
     • The probability of success ($\theta$) remains constant for every trial.
     • The process stops when the $k$-th success is achieved.

8. How does the negative binomial distribution differ from the binomial distribution?

   • Difference: In a binomial distribution, the number of trials ($n$) is fixed, and you calculate the probability of achieving a certain number of successes ($k$). In a negative binomial distribution, the number of successes ($k$) is fixed, and you calculate the probability of the number of trials ($x$) it takes to achieve those successes.

9. In what real-world scenarios can the negative binomial distribution be applied?

   • Applications: As mentioned in the conclusion, such as repeated attempts in sports, sales processes, troubleshooting, or any situation where the number of trials is variable and dependent on achieving a target number of successes.

10. How do you handle the probability of failure in the negative binomial formula?

    • Handling: The probability of failure is $(1 - \theta)$, where $\theta$ is the probability of success. It is raised to the power of $(x - k)$, representing the number of failures that occur in the $x$ trials to achieve the $k$ successes.