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) A local ice cream shop has decided to conduct a survey among its customers t...

Apr 25, 2024

) A local ice cream shop has decided to conduct a survey among its customers to find out how many of them prefer vanilla flavour over chocolate. The shopkeeper conducted the survey by asking 50 customers who visited the shop during peak hours. If 70% of the customers like vanilla, what is the probability that out of 10 customers, exactly 7 prefer vanilla (round your result to three decimal places)?

Solution by Steps

step 1

Calculate the probability of exactly 7 successes in 10 trials using the binomial probability formula:

P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}

where

n

is the number of trials,

k

is the number of successes,

p

is the probability of success, and

\binom{n}{k}

is the binomial coefficient

step 2

Given that

n=10

k=7

, and

p=0.7

, we find the binomial coefficient:

\binom{10}{7}

step 3

Calculate the binomial coefficient:

\binom{10}{7} = \frac{10!}{7!(10-7)!} = \frac{10!}{7!3!}

step 4

Simplify the binomial coefficient:

\binom{10}{7} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120

step 5

Calculate the probability of exactly 7 successes:

P(X=7) = 120 \times (0.7)^7 \times (0.3)^3

step 6

Simplify the probability:

P(X=7) = 120 \times 0.0823543 \times 0.027 = 0.266828

step 7

Round the result to three decimal places:

P(X=7) \approx 0.267

Answer

P(X=7) \approx 0.267

Key Concept

Binomial Probability Formula

Explanation

The binomial probability formula is used to calculate the probability of a given number of successes in a fixed number of independent trials, each with the same probability of success. In this case, it was used to find the probability of exactly 7 customers preferring vanilla out of 10, given a 70% chance that any one customer prefers vanilla.