Question

Round answers to 6 decimal places or type exact answers. A baker blends 555 raisins into a dough and makes 415 cookies. The probability of a raisin being in a randomly picked cookie is 1/415. Use the Poisson distribution to compute the following probabilities. What is the probability that a randomly picked cookie will have no raisins? What is the probability that a randomly picked cookie will have exactly 1 raisin? What is the probability that a randomly picked cookie will have exactly 3 raisins? What is the probability that a randomly chosen cookie will have at least two raisins in it?

Answer 1

To calculate the probabilities, use the Poisson distribution formula: P(X=k) = (e^(-λ) * λ^k) / k!. For no raisins, substitute k=0 and λ=1/415. For 1 raisin, substitute k=1 and λ=1/415. For exactly 3 raisins, substitute k=3 and λ=1/415. For at least 2 raisins, subtract the probabilities of 0 and 1 raisins from 1.

Step-by-step explanation:

To calculate the probabilities using the Poisson distribution, we will use the formula P(X=k) = (e^(-λ) * λ^k) / k!

a) To find the probability that a randomly picked cookie will have no raisins, we substitute k=0 and λ=1/415 into the formula. P(X=0) = (e^(-1/415) * (1/415)^0) / 0! = 0.9975

b) To find the probability that a randomly picked cookie will have exactly 1 raisin, we substitute k=1 and λ=1/415 into the formula. P(X=1) = (e^(-1/415) * (1/415)^1) / 1! = 0.0024

c) To find the probability that a randomly picked cookie will have exactly 3 raisins, we substitute k=3 and λ=1/415 into the formula. P(X=3) = (e^(-1/415) * (1/415)^3) / 3! = 0.0000003044

d) To find the probability that a randomly chosen cookie will have at least two raisins, we calculate the probability of having 0 or 1 raisins and subtract it from 1. P(X≥2) = 1 - [P(X=0) + P(X=1)]

= 1 - (0.9975 + 0.0024)

= 0.0001