Solutions - 5110-06

(1) Set up conditional probability formula:

P [X = n + k | X > n] = \frac{P [X = n + k \cap X > n]}{P [X > n]} = \frac{P [X = n + k]}{P [X > n]}

(2) Find formulas for numerator and denominator:

P [X = n + k] = p {(1 - p)}^{n + k - 1}

P [X > n] = \sum_{x = n + 1}^{\infty} {(1 - p)}^{x - 1} p = \frac{p {(1 - p)}^{n}}{1 - (1 - p)} = {(1 - p)}^{n}

(3) Plug in and simplify:

P [X = n + k | X > n] = \frac{p {(1 - p)}^{n + k - 1}}{{(1 - p)}^{n}} = p {(1 - p)}^{k - 1} = P [X = k]

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