Question

9. Discrete probability distributions #1

A study conducted by three law school professors found that asylum seekers in the United States face broad disparities in the nation’s immigration courts. The professors discovered that 54% of refugees who ask for asylum in the San Francisco immigration court win asylum, but only 12% are granted asylum in the Atlanta immigration court. [Source: Julia Preston, “Wide Disparities Found in Judging of Asylum Cases,” The New York Times, May 31, 2007.]

Select the appropriate distribution in the Distributions tool to help answer the questions that follow.

0123BinomialPoisson

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You randomly select 20 refugees who are asking for asylum in the San Francisco immigration court. Let X denote the number of asylum seekers who win their cases.

The probability that exactly 11 asylum seekers are granted asylum is   .

The probability that at least seven asylum seekers are granted asylum is   .

The expected value of X is   , and the standard deviation of X is

10. Discrete probability distributions #2

The Geminids is an annual meteor shower that appears every December. Under a clear, dark sky, an observer of the Geminids would see an average of 20 meteors per 10-minute period (if the meteors’ emanation point were directly overhead).

Select the appropriate distribution in the tool to help answer the following questions. (Note: You will need to read the questions first to determine the appropriate distribution.)

0123BinomialPoisson

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It’s December and you host a Geminids party on the peak night of the meteor shower. The sky is clear and dark, and the meteors’ emanation point is directly overhead. You and your friends watch the sky for 10 minutes. The probability that you see exactly 20 meteors is0.0888   .

The probability that you see more than 16 meteors while watching the night sky for 10 minutes is   .

Select the appropriate distribution in the tool below to help answer the following questions. (Note: You will need to read the questions first to determine the appropriate distribution.)

0123BinomialPoisson

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After going inside for a midnight snack, you and your friends go back outdoors for a 25-minute sky-gazing session. The probability that you observe no more than 48 meteors during this sky-gazing session is   .

The number of meteor sightings over 20 minutes has an expected value of    and a standard deviation of   

Answer 1

Q9: Binomial distribution

Here X follows a Binomial distribution with n=20 and p=0.54

Probability that exactly 11 asylum seekers are granted asylum = P(Y=11) = 7.77*10^-9. Here Y denote the number of asylum seekers who are granted asylum. then Y follows a binomial distribution with n=20 and p=0.54*0.12

The probability that at least seven asylum seekers are granted asylum = P(Y7) = 0.99998

The expected value of X is 20*0.54 = 10.8, and the standard deviation of X = =2.229

Q10: Poisson distribution

Let X denote the number of meteors seen while watching the night sky for 10minutes, Then x follows a Poisson distribution with parameter 20.

The probability that you see more than 16 meteors while watching the night sky for 10 minutes = P(X>16) = 0.7789

Let Y denote the number of meteors seen while watching the night sky for 25minutes, Then Y follows a Poisson distribution with parameter 50.

Thus the probability that you observe no more than 48 meteors during this sky-gazing session = P(Y48) = 0.42487

Let Y denote the number of meteors seen while watching the night sky for 20minutes, Then Y follows a Poisson distribution with parameters 40.

Thus E(Z) = 40 and standard deviation of Z is = 6.325