SOLUTION: Suppose that a particular physician determines that based on his experience with certain cancers, only 20% survive as long as 4 months following diagnosis. What is the probability

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Question 1018774: Suppose that a particular physician determines that based on his experience with certain cancers, only 20% survive as long as 4 months following diagnosis. What is the probability that 2 out of 10 patients survive the initial 4 months following diagnosis? Show your work.
B (k;n;p) =
Answer by mathmate(429) About Me  (Show Source):
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Question:
Suppose that a particular physician determines that based on his experience with certain cancers, only 20% survive as long as 4 months following diagnosis. What is the probability that 2 out of 10 patients survive the initial 4 months following diagnosis? Show your work.
B (k;n;p) =

Solution:
Binomial distribution is applicable if all of the following conditions are met:
1. Each trial is Bernoulli (success or failure), random, and independent of each other.
2. The number of trials is fixed and known.
3. The probability of success is known and remains constant throughout the trials.

For the given problem, all the conditions are met, so the binomial distribution is applicable.

The probability of k successes out of n trials each with a probability of success p is given by
B(k;n;p)=C%28n%2Ck%29%2Ap%5Ek%2A%281-p%29%5E%28n-p%29
where C(n,k)=n%21%2F%28k%21%2A%28n-k%29%21%29

with
p=0.2; k=2; n=10;
P(2;10;0.2)=C%2810%2C2%29%2A0.2%5E2%2A%280.8%29%5E8=0.302(approximately)

Answer: The probability of exactly 2 out of 10 patients survive the initial four months is 0.302.