It has been estimated that about 30% of frozen chicken contain enough salmonella bacteria to cause illness if improperly cooked. A consumer purchases 12 frozen chickens. What is the probability that the consumer will have more than 6 contaminated chickens? 0.882 0.961 0,039 0.079 0.118
The Correct Answer and Explanation is:
We are given a scenario where 30% of frozen chickens contain enough salmonella bacteria to cause illness, and the consumer purchases 12 chickens. We need to calculate the probability that more than 6 of the chickens are contaminated.
Problem Breakdown
This is a binomial probability problem, where the number of trials is n=12n = 12n=12 (the number of chickens), and the probability of success (a chicken being contaminated) is p=0.30p = 0.30p=0.30.
The probability of more than 6 contaminated chickens is the sum of the probabilities of having 7, 8, 9, 10, 11, or 12 contaminated chickens.
The binomial probability formula is:P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1 – p)^{n-k}P(X=k)=(kn)pk(1−p)n−k
where:
- n=12n = 12n=12 (the number of trials),
- p=0.30p = 0.30p=0.30 (the probability of contamination),
- kkk is the number of contaminated chickens,
- (nk)\binom{n}{k}(kn) is the binomial coefficient, or “n choose k.”
We want to find the probability of having more than 6 contaminated chickens, so we need to calculate:P(X>6)=1−P(X≤6)P(X > 6) = 1 – P(X \leq 6)P(X>6)=1−P(X≤6)
This is the complement of the probability of having 6 or fewer contaminated chickens.
Calculation
First, calculate P(X≤6)P(X \leq 6)P(X≤6), which is the cumulative probability for k=0k = 0k=0 to k=6k = 6k=6. The total probability is the sum of individual probabilities for each value of kkk from 0 to 6.
Using the binomial distribution:P(X=0)=(120)(0.30)0(0.70)12P(X = 0) = \binom{12}{0} (0.30)^0 (0.70)^{12}P(X=0)=(012)(0.30)0(0.70)12P(X=1)=(121)(0.30)1(0.70)11P(X = 1) = \binom{12}{1} (0.30)^1 (0.70)^{11}P(X=1)=(112)(0.30)1(0.70)11
… and so on up to k=6k = 6k=6.
Once you’ve calculated P(X≤6)P(X \leq 6)P(X≤6), subtract it from 1 to get P(X>6)P(X > 6)P(X>6).
Answer
Using a binomial probability calculator or software, the result of P(X>6)P(X > 6)P(X>6) is approximately:P(X>6)≈0.118P(X > 6) \approx 0.118P(X>6)≈0.118
Thus, the correct answer is 0.118.
This means there’s an 11.8% chance that more than 6 out of the 12 chickens will be contaminated with salmonella.
