In a bank an average of 5 out of every 10 customers ask for opening a new account. A random sample of 20 customers is selected. Calculate the probability that exact 7 customers ask for opening a new account. Round your answer four decimal places.
Question
In a bank an average of 5 out of every 10 customers ask for opening a new account. A random sample of 20 customers is selected. Calculate the probability that exact 7 customers ask for opening a new account. Round your answer four decimal places.
Solution
This is a binomial probability problem. The formula for binomial probability is:
P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
where:
- P(X=k) is the probability of k successes in n trials
- C(n, k) is the number of combinations of n items taken k at a time
- p is the probability of success on a single trial
- n is the number of trials
- k is the number of successes
In this case:
- n = 20 (the number of customers)
- k = 7 (the number of customers who ask for opening a new account)
- p = 0.5 (the probability of a customer asking for opening a new account)
Substituting these values into the formula, we get:
P(X=7) = C(20, 7) * (0.5^7) * ((1-0.5)^(20-7))
Calculating the above expression gives a probability of approximately 0.2026 when rounded to four decimal places.
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