Probability in Binomial Distribution: Calculating $P(X=0)$

Binomial Distribution & Discrete Random Variables

Calculating the Probability of Zero Successes

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success. The probability of observing exactly k successes in n trials is given by the binomial probability formula:

• Formula:

$P(X = k) = {n \choose k} p^k (1-p)^{n-k}$

• Where:

  • n = number of trials

  • k = number of successes

  • p = probability of success on a single trial

  • 1-p = probability of failure on a single trial

Example: Probability of Zero Successes

Suppose the probability of failure on a single trial is 0.85, and there are 10 independent trials. The probability of observing zero successes (i.e., all trials are failures) is:

$P(X = 0) = 0.85^{10} = 0.1969$

• Interpretation: There is approximately a 19.69% chance that all 10 trials result in failure (zero successes), given the probability of failure per trial is 0.85.

Additional info: This calculation is a special case of the binomial distribution where k = 0. The binomial coefficient ${n \choose 0} = 1$, so the formula simplifies to $P(X=0) = (1-p)^n$.