SOLUTION: The Center for Disease Control, or CDC, has determined that when a person is given a vaccine, the probability that the person will develop immunity to the virus is 0.8. If 4 people

Question 1086299: The Center for Disease Control, or CDC, has determined that when a person is given a vaccine, the probability that the person will develop immunity to the virus is 0.8. If 4 people are given this vaccine, find the probability that at least 1 of the people will develop immunity to the virus. Give your answer correct to 4 decimal places?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Define "success" to mean "developing immunity"
X = random discrete variable counting number of successes (takes on the values from X = 0 to X = 4)
n = 4 is the sample size
p = 0.8 is the probability of success (probability of developing immunity)

If we can find P(X = 0), then we can use it to find P(X > 0). Note how P(X = 0) + P(X > 0) = 1. So P(X > 0) = 1 - P(X = 0)

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In this case, k = 0. Also use n = 4 and p = 0.8

First find the combination value
n C k = (n!)/(k!*(n-k)!)
4 C 0 = (4!)/(0!*(4-0)!)
4 C 0 = (4!)/(0!*4!)
4 C 0 = (1)/(0!*1)
4 C 0 = (1)/(1)
4 C 0 = 1

Then use it to compute the binomial probability value
P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)
P(X = 0) = (4 C 0)*(0.8)^(0)*(1-0.8)^(4-0)
P(X = 0) = (4 C 0)*(0.8)^(0)*(0.2)^(4)
P(X = 0) = (1)*(0.8)^(0)*(0.2)^4
P(X = 0) = (1)*(1)*(0.0016)
P(X = 0) = 0.0016

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We know the value of P(X = 0) now. We can use it to find P(X > 0)

P(X > 0) = 1 - P(X = 0)
P(X > 0) = 1 - 0.0016
P(X > 0) = 0.9984

Because P(X >= 1) = P(X > 0), we have found the answer.
The probability that at least one person develops immunity is 0.9984

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