In a large restaurant, an average of 2 out of every 5 customers ask for water with their meal. A random sample of 10 customers is selected.Calculate the probability taht exact 8 customers ask for water with their meal. Enter your answer as a decimal rounded to 4 decimal places
Question
In a large restaurant, an average of 2 out of every 5 customers ask for water with their meal. A random sample of 10 customers is selected.Calculate the probability taht exact 8 customers ask for water with their meal. Enter your answer as a decimal rounded to 4 decimal places
Solution
This is a binomial probability problem. The binomial probability formula 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 combination 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 = 10 (the number of customers)
- k = 8 (the number of customers who ask for water)
- p = 2/5 (the probability that a customer asks for water)
Step 1: Calculate C(n, k) C(n, k) = n! / [k!(n-k)!] C(10, 8) = 10! / [8!(10-8)!] = 45
Step 2: Calculate p^k (2/5)^8 = 0.00256
Step 3: Calculate (1-p)^(n-k) (1 - 2/5)^(10-8) = (3/5)^2 = 0.36
Step 4: Multiply the results of steps 1, 2, and 3 to find the probability P(X=8) = C(n, k) * (p^k) * ((1-p)^(n-k)) P(X=8) = 45 * 0.00256 * 0.36 = 0.041472
So, the probability that exactly 8 out of 10 randomly selected customers ask for water with their meal is approximately 0.0415 when rounded to four decimal places.
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