SOLUTION: 38% of college students say they use credit cards because of the reward program. you randomly select 10 college students and ask each the reason he or she uses credit cards. Find t

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Question 939754: 38% of college students say they use credit cards because of the reward program. you randomly select 10 college students and ask each the reason he or she uses credit cards. Find the probabilty that the number of college students who say they use credit cards because of rewards program is (a) exactly two (b) more than two (c) between two and five inclusive
Answer by mathmate(429) About Me  (Show Source):
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Given:
a n=10 step Bernoulli experiment (exactly two possible outcomes).
probability is constant at p=38%.
each step of the experiment is independent of the other.

Under the given conditions, the problem can be solved using the binomial distribution:
P(X=r)=C%28n%2Cr%29%2Ap%5Er%2A%281-p%29%5E%28n-r%29+
where
r is the number of successes (use credit card because of reward).
n = 10, number of students interviewed in the experiments
p = 0.38, probability of success
C(n,r) = n%21%2F%28%28n-r%29%21r%21%29combination of r objects out of n

(a) exactly 2 successes (out of ten)
P(X=2) = C%2810%2C2%29%2A0.38%5E2%2A%281-0.38%29%5E8+=+0.1419
Answer: probability of exactly 2 successes is

(b) more than 2 successes (i.e. 3 to 10)
It is easier to calculate 1-(P(X=0)+P(X=1)+P(X=2) which gives the same answer.
P(X=0) = C%2810%2C0%29%2A0.38%5E0%2A%281-0.38%29%5E10+=+0.0084
P(X=1) = C%2810%2C1%29%2A0.38%5E1%2A%281-0.38%29%5E9+=+0.0514
P(X=2) = C%2810%2C2%29%2A0.38%5E2%2A%281-0.38%29%5E8+=+0.1419
P(X>2) = 1-(P(X=0)+P(X=1)+P(X=2) = 0.0084%2B0.0514%2B0.1419 = 0.2017
Answer: probability of greater than 2 successes is 0.2017

(c) between two and five
P(X=2) = C%2810%2C2%29%2A0.38%5E2%2A%281-0.38%29%5E8+=+0.1419
P(X=3) = C%2810%2C3%29%2A0.38%5E3%2A%281-0.38%29%5E7+=+0.2319
P(X=4) = C%2810%2C4%29%2A0.38%5E4%2A%281-0.38%29%5E6+=+0.2487
P(X=5) = C%2810%2C5%29%2A0.38%5E5%2A%281-0.38%29%5E5+=+0.1829
P(2<= X <= 5) = P(X=2)+P(X=3)+P(X=4)+P(x=5)=0.1419%2B0.2319%2B0.2487%2B0.1829
=0.8054
Answer: probability of between 2 to 5 successes is 0.8054