A camera manufacturer just shipped a batch of 36 digital cameras to local retailers and seven of the cameras have a defective sensor. Cutler Electronics received eight of these digital cameras. What is the probability that two of the cameras shipped to Cutler are defective?
A)0.2155
B)0.3297
C)0.0516
D)0.1374

Question

Quizplus · Accepted Answer

To find the probability that two of the cameras shipped to Cutler are defective, we can use the hypergeometric distribution formula, which is appropriate for problems of this type where we are dealing with successes (defective cameras) and failures (non-defective cameras) without replacement. The formula for the hypergeometric distribution is:$$ P(X=k) = \frac{{\binom{K}{k} \cdot \binom{N-K}{n-k}}}{{\binom{N}{n}}} $$Where:- $N$ is the total number of items (36 cameras),- $K$ is the total number of successes in the population (7 defective cameras),- $n$ is the number of items drawn (8 cameras received by Cutler),- $k$ is the number of successes in the drawn sample (2 defective cameras we are calculating the probability for),- $\binom{a}{b}$ is the binomial coefficient, representing the number of ways to choose $b$ successes out of $a$ total items.Plugging in the numbers:$$ P(X=2) = \frac{{\binom{7}{2} \cdot \binom{29}{6}}}{{\binom{36}{8}}} $$$$ P(X=2) = \frac{{21 \cdot 475020}}{{90751320}} $$$$ P(X=2) \approx 0.3297 $$Therefore, the probability that two of the cameras shipped to Cutler are defective is approximately 0.3297.

A Camera Manufacturer Just Shipped a Batch of 36 Digital