Probability & Statistics Ch.2 Section 3 Question 19

· Mohammad-Ali Bandzar

Solutions for “Probability and Statistics: The Science of Uncertainty” (Second Edition). These are solutions I have come up with; I offer no guarantee of accuracy.

Question

Suppose an urn contains 1000 balls — one of these is black, and the other 999 are white. Suppose that 100 balls are randomly drawn from the urn with replacement. Use the appropriate Poisson distribution to approximate the probability that five black balls are observed.

Solution

From Example 2.3.6 in the textbook:

pY(y)=P(Y=y)=λyy!e−λp_Y(y) = P(Y=y) = \frac{\lambda^y}{y!}e^{-\lambda}

For Poisson approximation to binomial:

  • λ = (number of draws) × (probability of drawing a black ball)
  • λ = 100 × (1/1000) = 0.1

y = 5 (desired number of black balls)

P(Y=5)=0.155!e−0.1P(Y=5) = \frac{0.1^5}{5!}e^{-0.1}

=0.00001120×0.9048= \frac{0.00001}{120} \times 0.9048

P(Y=5)≈7.54×10−8P(Y=5) \approx 7.54 \times 10^{-8}