A salesperson goes door-to-door in a residential area to demonstrate the use of a new Household appliance to potential customers. She has found from her years of experience that after demonstration, the probability of purchase (long run average) is 0.30. To perform satisfactory on the job, the salesperson needs at least four orders this week. If she performs 15 demonstrations this week, what is the probability of her being satisfactory? What is the probability of between 4 and 8 (inclusive) orders?
Now the challenging questions: if the salesperson wants to be at least 90 percent confident of getting satisfactory evaluation in her job this week, how many demonstrations should she perform? How would your answers to above questions change if the probability of success increases (say by training) to 0.35?
To solve these probability questions, we need to use the binomial distribution formula. The binomial distribution is used when there are two possible outcomes (success or failure) for each trial, and each trial is independent.
1. Probability of being satisfactory:
The salesperson needs at least four orders, which means she needs to have four or more successful demonstrations out of the 15. We'll calculate the probability of getting exactly four, five, six, seven, eight, up to fifteen successful demonstrations, and add them up to find the probability of being satisfactory.
P(satisfactory) = P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8) + ... + P(X=15)
2. Probability of between 4 and 8 (inclusive) orders:
To find the probability of between 4 and 8 orders, we can sum up the probabilities of getting four, five, six, seven, and eight successful demonstrations.
P(4 ≤ X ≤ 8) = P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8)
Now let's calculate these probabilities.
3. Number of demonstrations needed for at least 90% confidence:
To find the number of demonstrations needed for at least 90% confidence of getting a satisfactory evaluation, we need to find the minimum number of demonstrations that will achieve a cumulative probability of at least 0.90 for getting four or more successful demonstrations. We will iterate through increasing numbers of demonstrations until we reach a cumulative probability of at least 0.90.
4. Impact of increased probability of success:
If the probability of success increases to 0.35, then we need to recalculate all the probabilities mentioned above (probability of being satisfactory, probability of between 4 and 8 orders, and number of demonstrations needed for 90% confidence).
Now let's calculate each of these probabilities and answer the questions.