Answer:
To find the probability that thirty-eight or fewer of the 50 new products introduced in the grocery store will fail, you can use the binomial probability formula. In this case, you want to calculate the cumulative probability for a binomial distribution.
The binomial probability formula is:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting exactly k successes.
n is the number of trials (in this case, the number of new products introduced, which is 50).
k is the number of successes (in this case, the number of products that fail).
p is the probability of success on a single trial (in this case, the probability that a new product fails).
(n choose k) is the binomial coefficient, calculated as C(n, k) = n! / (k! * (n - k)!), which represents the number of ways to choose k successes out of n trials.
In this problem, p (the probability that a new product fails) is 0.73 because 73% of new products fail, so p = 0.73. We want to find the probability that thirty-eight or fewer of the 50 new products fail, which means we need to calculate the cumulative probability for k ranging from 0 to 38.
Now, calculate the probability for each value of k and sum them up:
P(X ≤ 38) = Σ [P(X = k) for k = 0 to 38]
P(X ≤ 38) = Σ [(50 choose k) * (0.73^k) * (0.27^(50 - k)) for k = 0 to 38]
You can calculate this sum using a calculator or a statistical software package. It's a cumulative binomial distribution.
However, if you prefer not to calculate it manually, you can use statistical software or an online binomial calculator to find the cumulative probability. The result will be the probability that thirty-eight or fewer of the 50 new products fail.
Explanation: