it has been estimated that about 30% of frozen chickens contain enough salmonella bacteria to cause illness if imprperly cooked. A consumer purchases 12 frozen chickens. What is the probability that the consumer will have more than 6 contaminated chickens? A) .882 B) .961 C) .039 D) .079
The Correct Answer and Explanation is :
The correct answer is C) 0.039.
Explanation:
This problem follows a binomial probability distribution since each frozen chicken can either be contaminated or not. The probability of contamination per chicken is 30% (0.30), and the consumer purchases 12 chickens. We define:
- ( n = 12 ) (total chickens)
- ( p = 0.30 ) (probability of contamination per chicken)
- ( X ) is the number of contaminated chickens, which follows a binomial distribution:
[
X \sim Binomial(n=12, p=0.30)
] - We need to find ( P(X > 6) ), i.e., the probability that more than 6 chickens are contaminated.
Using the cumulative distribution function (CDF) of the binomial distribution, we compute:
[
P(X > 6) = 1 – P(X \leq 6)
]
Using statistical calculations, ( P(X \leq 6) ) is approximately 0.9614, so:
[
P(X > 6) = 1 – 0.9614 = 0.0386
]
Rounding to three decimal places gives 0.039, which matches option C.
Interpretation:
This result tells us that there is only about a 3.9% chance that more than 6 out of 12 frozen chickens will be contaminated. While contamination is possible, having a majority of the chickens (more than half) contaminated is relatively unlikely.
Now, I’ll generate an image illustrating the binomial probability distribution for this scenario.
The bar chart above visualizes the binomial probability distribution for this problem. The red bars highlight the probability of having more than 6 contaminated chickens (i.e., ( P(X > 6) )). This confirms that the probability is small, around 0.039 (3.9%).
Let me know if you need further clarification!
