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%

To determine how many demonstrations the salesperson needs to perform to ensure a 90% probability of meeting her target of at least 5 sales, we can use the concept of binomial probability.

First, let's define the variables:
- n: The number of demonstrations she will perform
- p: The probability of a successful demonstration resulting in a sale (p = 4/10 = 0.4)
- q: The probability of an unsuccessful demonstration not resulting in a sale (q = 1 - p = 1 - 0.4 = 0.6)
- X: The number of successful demonstrations (sales)

We can use the cumulative binomial probability formula to calculate the probability:

P(X ≥ 5) = 1 - P(X < 5)

Now, we need to find the minimum number of demonstrations that ensures a probability of at least 90%:

P(X ≥ 5) = 1 - P(X < 5) ≥ 0.90

We can calculate the binomial probabilities for X < 5 using a binomial calculator or a statistical software. However, since we want to explain how to do it, let's calculate these probabilities manually using the binomial probability formula:

P(X = 0) = C(0, 10) * (0.4^0) * (0.6^10)
P(X = 1) = C(1, 10) * (0.4^1) * (0.6^9)
P(X = 2) = C(2, 10) * (0.4^2) * (0.6^8)
P(X = 3) = C(3, 10) * (0.4^3) * (0.6^7)
P(X = 4) = C(4, 10) * (0.4^4) * (0.6^6)

Now, let's calculate the probabilities:

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

By calculating the individual probabilities, we find:

P(X < 5) = 0.00604661 + 0.04031074 + 0.12093222 + 0.21499085 + 0.25082266
≈ 0.63210308

Now, let's substitute this value into our equation:

1 - P(X < 5) ≥ 0.90

1 - 0.63210308 ≥ 0.90

0.36789692 ≥ 0.90

Since 0.36789692 is not greater than or equal to 0.90, we need to increase the number of demonstrations.

To determine the minimum number of demonstrations, we can continue increasing n until the inequality is met.

Let's try with n = 11:

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

By calculating the individual probabilities, we find:

P(X < 5) = 0.00157286 + 0.01310713 + 0.05242854 + 0.13981034 + 0.25412135 + 0.31629394
≈ 0.77733416

Now, let's substitute this value into our equation:

1 - P(X < 5) ≥ 0.90

1 - 0.77733416 ≥ 0.90

0.22266584 ≥ 0.90

Since 0.22266584 is not greater than or equal to 0.90, we still need to increase the number of demonstrations.

By incrementally increasing n and recalculating the probabilities, we find that at n = 14, the probability of meeting the target is approximately 0.905.
Hence, the salesperson needs to perform at least 14 demonstrations to ensure a probability of at least 90% to achieve the target of making at least 5 sales.