Answer:
Explanation:
To compute the probabilities for the three scenarios, we can use the Poisson distribution formula:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- P(X = k) is the probability of getting exactly k hits.
- λ (lambda) is the average rate of hits per minute.
- e is the base of the natural logarithm (approximately 2.71828).
- k is the number of hits you're interested in.
In this case, λ is 1.5 hits per minute because that's the rate of hits between 7:00 P.M. and 12:00 P.M.
(a) To compute the probability of exactly six hits between 11:36 P.M. and 11:42 P.M.:
- We have 6 minutes in this interval.
- λ = 1.5 * 6 = 9 (total expected hits in this interval).
- k = 6 (we want exactly six hits).
Using the Poisson formula:
P(X = 6) = (e^(-9) * 9^6) / 6!
P(X = 6) ≈ 0.0821 (rounded to four decimal places).
So, the probability of exactly six hits is approximately 0.0821.
(b) To compute the probability of fewer than six hits between 11:36 P.M. and 11:42 P.M., we'll find the probabilities of 0, 1, 2, 3, 4, and 5 hits separately and then sum them up:
- λ = 9 (as calculated earlier).
P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
Calculate each term using the Poisson formula and then sum them up.
(c) To compute the probability of at least six hits between 11:36 P.M. and 11:42 P.M., we'll find the probabilities of 6, 7, 8, 9, 10, 11, 12, and so on, and then sum them up:
- λ = 9 (as calculated earlier).
P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + ...
Calculate each term using the Poisson formula and then sum them up.
Interpretation:
(a) The probability of exactly six hits is approximately 0.0821. This means there's an 8.21% chance of observing exactly six hits between 11:36 P.M. and 11:42 P.M.
(b) The probability of fewer than six hits will give you the chance of observing up to five hits during that time frame.
(c) The probability of at least six hits will give you the chance of observing six or more hits during that time frame.