Exam 1                        MAT 221                                NAME:

 

Directions: Read each question carefully. Clearly label your answers! Show all work in blue books. Keep your calculators and formula sheets to yourself! Round answers to 4 decimal places where applicable!

 

  1. On the Island of Bob, in the Caribbean, there is a 20% chance of rain each day. The weather is independent from day to day. You go to the Island of Bob for 4 days:

a.                   What is the probability it rains at least 1 of the 4 days?

b.                  What is the probability it rains all 4 days?

c.                   What is the probability it rains none of the 4 days?

 

  1. Dr. Hibbard develops a new test for rabies, which is known to be present in 2% of the population of Springfield. The test is positive 99% of the time given the patient has rabies. The test is negative 95% of the time given the patient does NOT have rabies.

a.                   What is the probability a random patient tests positive AND has rabies?

b.                  Given the test is positive, what is the probability the patient does have rabies?

c.                   Given the test is negative, what is the probability the patient does not have rabies?

 

  1. Lisa’s tests scores for her advanced math class are: 92, 97, 100, 79, and 97.

a.                   Find the mean, median, and mode of her scores.

b.                  If the standard deviation of her scores is 8.34, are any of her scores abnormal? JUSTIFY!

 

  1. From a batch of 24 cellular phones, 5 are faulty. A random sample of 3 phones is taken.

a.                   Find the probability none of the 3 phones is faulty.

b.                  Find the probability all 3 phones are faulty.

 

  1. The probability that Marge wins first place in a raffle is 0.32. The probability that she wins second place is 0.47. Find the probability she wins either first or second place or both, if:

a.                   winning first place and winning second place are mutually exclusive events.

b.                  winning first place and winning second place are independent events.

c.                   the probability she wins first place given she wins second place is 0.10.

Note : for this problem you need to find 1 probability for each part.

 

Answers:

 

1.         a.         P(No Rain) = .8 ^ 4 = .4096.   P(1 or more Rain) = 1 - .4096 = .5904

            b.         P(4 Rain) = .2 ^ 4 = .0016

            c.         P(No Rain) = .4096

 

2.         P(R) = .02        P(T | R) = .99   P(T’ | R’) = .95

            a.         P(R ∩ T) = .99 * .02 = .0198

            b.         P( R | T ) = .99 * .02 / (.99 * .02 + .05 * .98) = .2878

            c.         P(R’ | T’) = .95 * .98 / (.95 * .98 + .01 * .02) = .9998

 

3.         mode = 97, mean = 93, median = 97, 79 is only 1.68 standard deviation less than the mean and            therefore is NOT abnormal.

 

4.         P(None faulty) = 5 C 0 * 19 C 3 / 24 C 3 = 969 / 2024 = .4787

            P(All faulty ) = 5 C 3 * 19 C 0 = 10 / 2024 = .0049

 

5.         A = {wins 1st place}     B = {wins 2cd place}

            a.         P( A U B) = .32 + .47 – 0 = .79

            b.         P( A U B) = .32 + .47 - .1504 = .6396

            c.         P( A U B) = .32 + .47 - .047 = .743