Astrid is looking for a job at a call center. Call center A offers her 15 dollars per hour and call center B offers her 25 cents
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Answer:
(A) The probability that the number of drivers will be at most 10 is 0.0108.
(B) The probability that the number of drivers will exceed 20 is 0.4409.
(C) The probability that the number of drivers will be between 10 and 20 (inclusive) is 0.5483 and excluding 10 and 20 is 0.4595.
(D) The probability that the number of drivers will within 2 standard deviations of the mean is 0.9549.
Step-by-step explanation:
Let <em>X </em>= number of drivers who travel a distance during a designated time period.
The random variable <em>X</em> follows a Poisson distribution with parameter <em>λ</em> = 20.
The probability function of a Poisson distribution is:
(A)
Compute the probability that the number of drivers will be at most 10 as follows:
P (X ≤ 10) = P (X = 0) + P (X = 1) + P (X = 2) ... + P (X = 10)
Thus, the probability that the number of drivers will be at most 10 is 0.0108.
(B)
Compute the probability that the number of drivers will exceed 20 as follows:
P (X > 20) = 1 - P (X ≤ 20)
= 1 - [P (X = 0) + P (X = 1) + P (X = 2) ... + P (X = 19) + P (X = 20)]
Thus, the probability that the number of drivers will exceed 20 is 0.4409.
(C)
Compute the probability that the number of drivers will be between 10 and 20 (inclusive) as follows:
P (10 ≤ X ≤ 20) = P (X ≤ 20) - P (X ≤ 10)
Compute the probability that the number of drivers will be exactly between 10 and 20 as follows:
P (10 < X < 20) = P (X = 11) + P (X = 12) + ... + P (X = 19)
Thus, the probability that the number of drivers will be between 10 and 20 (inclusive) is 0.5483 and excluding 10 and 20 is 0.4595.
(D)
The mean of a Poisson distribution is,
The standard deviation of a Poisson distribution is:
Compute the probability that the number of drivers will within 2 standard deviations of the mean as follows:
Thus, the probability that the number of drivers will within 2 standard deviations of the mean is 0.9549.