On Statistics

The Negative Binomial Distribution: Counting Trials Until “r” Successes

In the realm of probability, understanding how many trials are needed until a specific number of successes occur is crucial. The negative binomial distribution emerges as a powerful tool in this context, focusing on calculating the probability of encountering r successes in a series of independent and identically distributed (i.i.d.) trials with two possible outcomes: success and failure.

What is the Negative Binomial Distribution?

The negative binomial distribution is a discrete probability distribution that describes the probability of having k failures before encountering the r-th success in a series of i.i.d. trials. Here’s what these terms mean:

  • Discrete: The random variable (number of failures before r successes) can only take on whole number values (0, 1, 2, …).
  • Independent trials: Each trial has no bearing on the outcome of other trials.
  • Identically distributed: The probability of success remains constant (p) across all trials.
  • r successes: This refers to the specific number of successes needed before the experiment stops.

What Does the Negative Binomial Distribution Tell Us?

This distribution allows us to calculate:

  • The probability of experiencing k failures before encountering the r-th success (P(X = k)).
  • The expected number of failures (trials) before encountering the r-th success (mean).
  • The probability of not having r successes within a specific number of trials.

Understanding Key Formulas

  1. Probability of k failures before the r-th success:

P(X = k) = (C(k + r - 1, r - 1) * p^r * (1 - p)^k)

  • X represents the random variable (number of failures before r successes).
  • k represents the number of failures.
  • p represents the probability of success in a single trial.
  • r represents the desired number of successes before the experiment stops.
  • C(x, y) represents the binomial coefficient, calculating the number of ways to choose x objects from a group of y objects.
  1. Expected number of failures before the r-th success (mean):

E(X) = q / p = (1 - p) / p

Interpreting the Formulas:

  • The binomial coefficient term accounts for the different combinations of successes and failures that can lead to encountering r successes after experiencing k failures.
  • The expected number (mean) indicates the average number of failures one can expect before encountering the r-th success.

Examples:

  1. Shooting for a Basketball Shot: You keep shooting baskets until you make 3 successful shots (r = 3). The probability of success (p) in each shot is 0.4. What is the probability of missing 2 shots (k = 2) before making your third successful shot?

P(X = 2) = (C(2 + 3 - 1, 3 - 1) * 0.4^3 * (1 - 0.4)^2) ≈ 0.2304

  1. Plant Germination: Planting seeds, you expect 5 seeds to germinate (r = 5) before stopping. The probability of successful germination (p) for each seed is 0.7. What is the expected number of seeds you need to plant before 5 germinate?

E(X) = (1 - 0.7) / 0.7 \approx 1.4286

Limitations of the Negative Binomial Distribution

  • It applies to situations with two possible outcomes (success and failure) in each trial.
  • It assumes independence and identical distribution of trials.

The negative binomial distribution serves as a valuable tool for understanding and analyzing the probability of encountering a specific number of successes within a series of trials. By grasping its concepts, formulas, and examples, you can delve into solving problems related to repeated trials and gain insights into the probability of achieving a desired outcome within a sequence of events.

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