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Negative binomial: 5th hit on 10th throw

Short answer

Negative binomial distribution gives the trial number of the k-th success. For 10th throw as 5th hit with p=0.6, probability=0.1003, mean=8.33, variance=5.56.

Define negative binomial Distribution. If a boy is throwing stone at a target what is the probability that his 10th throw is his 5th hit, if the probability of hitting the target at any trial is 0.6. Also find the mean and variance of random variable.

Negative binomial distribution

In independent Bernoulli trials with constant success probability p, let X be the trial number on which the k-th success occurs. Then X has a negative binomial distribution:

P(X = x) = C(x − 1, k − 1) pᵏ qˣ⁻ᵏ, x = k, k + 1, k + 2, …

Mean = k/p, variance = kq/p². (For k = 1 it reduces to the geometric distribution.)

Probability that the 10th throw is the 5th hit

Here k = 5, x = 10, p = 0.6, q = 0.4. The 10th throw must be a hit, and exactly 4 hits must occur in the first 9 throws.

P(X = 10) = C(9, 4) (0.6)⁵ (0.4)⁵ = 126 × 0.07776 × 0.01024 = 0.1003

Mean and variance

Mean = k/p = 5/0.6 = 8.33 throws

Variance = kq/p² = 5 × 0.4 / 0.36 = 2/0.36 = 5.56

Answer: probability = 0.1003, mean = 8.33, variance = 5.56

✓ Answer checked by Claude Sonnet 5.5

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