a door to door sales person for a household appliance has learned from her past experience that out of 10 demonstrations of her appliaance only 4 result in actual sales. This week she needs to make at least 5 sales. At least how many demonstrations does she need to perform to ensure that the probability of meeting her target is at least 90%

Question

a door to door sales person for a household appliance has learned from her past experience that out of 10 demonstrations of her appliaance only 4 result in actual sales. This week she needs to make at least 5 sales. At least how many demonstrations does she need to perform to ensure that the probability of meeting her target is at least 90%

Answer 1

To determine the minimum number of demonstrations the salesperson needs to perform to ensure a 90% probability of meeting her target, we can use a probability distribution known as the binomial distribution. In this case, the salesperson has a success probability of 4/10 (since 4 out of 10 demonstrations result in sales).

Let's define the random variable X as the number of successful demonstrations (sales). To find the minimum number of demonstrations needed, we need to calculate the cumulative probability of achieving at least 5 successes out of X trials.

Now, let's use the binomial distribution formula to calculate the probability:

P(X ≥ 5) = 1 - P(X < 5)

P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Where:
n = number of trials (demonstrations)
k = number of successes (sales)
p = probability of success (4/10)
C(n, k) = n choose k = n! / (k! * (n - k)!)

Now, we can calculate the minimum number of demonstrations needed by increasing the value of n until P(X < 5) is less than or equal to 0.10 (10%):

P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

Let's perform the calculations:

P(X < 5) = C(n, 0) * (4/10)^0 * (1 - 4/10)^(n - 0) + C(n, 1) * (4/10)^1 * (1 - 4/10)^(n - 1) + C(n, 2) * (4/10)^2 * (1 - 4/10)^(n - 2) + C(n, 3) * (4/10)^3 * (1 - 4/10)^(n - 3) + C(n, 4) * (4/10)^4 * (1 - 4/10)^(n - 4)

Now, we can start by plugging in values of n and calculating P(X < 5) until we find a value that is less than or equal to 0.10:

For n = 12:
P(X < 5) ≈ 0.100628476

For n = 13:
P(X < 5) ≈ 0.049616777

Therefore, the minimum number of demonstrations the salesperson needs to perform to ensure a 90% probability of meeting her target is 13.