Final answer:
Using the Poisson probability formula, we would calculate the expected number of cookies with an extra fortune (λ = 145 * 2%) and then use it to find the probabilities of 0, 1, 2, and 3 extra fortunes, totaling these probabilities and subtracting from 1 to find the probability of more than three having an extra fortune.
Step-by-step explanation:
The question involves using the Poisson distribution to solve for the probability of a specific number of events (in this case, fortune cookies with an extra fortune) occurring within a given sample size (145 fortune cookies). The student is specifically interested in the probability that more than three cookies contain an extra fortune. To solve this, we can use the formula for the Poisson probability:
P(X = x) = (e^{-λ} * λ^x) / x!
Where λ is the expected number of successes (extra fortunes) in the sample, which can be determined by multiplying the probability of success (2%) by the number of trials (145 cookies).
- Calculate λ: λ = 145 * 0.02 = 2.9
- Since we want the probability of more than 3 extra fortunes, we need to find 1 minus the probability of 0, 1, 2, and 3 extra fortunes.
To find the probability of more than three cookies having an extra fortune, we calculate:
P(X > 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]
Where each P(X = x) is calculated using the Poisson formula.