For the binomial distribution, (nx)𝑛𝑥px(1 − p)n − x, represents the probability of any particular sequence with x successes and n − x failures. Use this formula to answer the following: In the Southern area of the United States, approximately 20% of adults have a college degree. We randomly ask four adults whether they have a college degree. What is the probability that none of the adults have a college degree?Multiple choice question..512zero.16.4096
Question
For the binomial distribution, (nx)𝑛𝑥px(1 − p)n − x, represents the probability of any particular sequence with x successes and n − x failures. Use this formula to answer the following: In the Southern area of the United States, approximately 20% of adults have a college degree. We randomly ask four adults whether they have a college degree. What is the probability that none of the adults have a college degree?Multiple choice question..512zero.16.4096
Solution
To solve this problem, we can use the binomial distribution formula:
P(X=x) = (n choose x) * p^x * (1-p)^(n-x)
where:
- n is the number of trials (in this case, the number of adults we ask, which is 4)
- x is the number of successes we want (in this case, the number of adults with a college degree, which is 0 because we want to find the probability that none of the adults have a college degree)
- p is the probability of success on a single trial (in this case, the probability an adult has a college degree, which is 20% or 0.2)
- (n choose x) is a combination which calculates the number of possible combinations of n items taken x at a time.
Substituting the given values into the formula, we get:
P(X=0) = (4 choose 0) * (0.2)^0 * (1-0.2)^(4-0)
Calculating this gives:
P(X=0) = 1 * 1 * (0.8)^4 = 0.4096
So, the probability that none of the adults have a college degree is 0.4096.
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