Therese Felser manages a grocery warehouse which encourages volume shopping on the part of its customers. Therese has discovered that, on any given weekday 70 percent of the customer sales amount to more than $ 100. That is any given sale on such a day has a probability of 0.70 of being for more than $ 100.( Actually the conditional probabilities throughout the day would change slightly, depending on earlier sales, but this effect would be negligible for the first several sales of the day, so we can treat them as independent.) The first two sales on Wednesday are both for more than $ 100. None of the first three sales on Wednesday is for more than $ 100

Question

Therese Felser manages a grocery warehouse which encourages volume shopping on the part of its customers. Therese has discovered that, on any given weekday 70 percent of the customer sales amount to more than $ 100. That is any given sale on such a day has a probability of 0.70 of being for more than $ 100.( Actually the conditional probabilities throughout the day would change slightly, depending on earlier sales, but this effect would be negligible for the first several sales of the day, so we can treat them as independent.) The first two sales on Wednesday are both for more than $ 100. None of the first three sales on Wednesday is for more than $ 100

Answer 1

To answer this question, we need to calculate the probability of the fourth sale being more than $100 given that none of the first three sales were for more than $100.

Let's break down the information given:

- On any given weekday, there is a probability of 0.70 for a sale to be more than $100.
- The first two sales on Wednesday are both for more than $100.
- None of the first three sales on Wednesday is for more than $100.

We want to find the probability of the fourth sale being more than $100, given that the first three sales were all less than or equal to $100.

To calculate this probability, we need to understand conditional probability. The probability of event A given event B is denoted as P(A|B) and is calculated as:

P(A|B) = P(A ∩ B) / P(B)

In this case, event A is the fourth sale being more than $100, and event B is that the first three sales were all less than or equal to $100.

To calculate P(A ∩ B), we need to consider the probability of both event A and event B happening. Since the sales are treated as independent, we can multiply the probabilities:

P(A ∩ B) = P(A) * P(B)

Given that the probability of any given sale being more than $100 is 0.70, we can say:

P(A ∩ B) = 0.70 * 0.70 = 0.49

Now, to calculate P(B), which is the probability that the first three sales were all less than or equal to $100, we need to find the complement of the probability that they were more than $100.

P(B) = 1 - P(all sales > $100)

Since the first two sales were more than $100, the probability of the third sale being less than or equal to $100 is 1, since it is certain.

Therefore:

P(B) = 1 - P(all sales > $100) = 1 - 0.70 = 0.30

Now we can calculate P(A|B) using the formula mentioned earlier:

P(A|B) = P(A ∩ B) / P(B) = 0.49 / 0.30 ≈ 0.1633

So, the probability of the fourth sale being more than $100, given that the first three sales were all less than or equal to $100, is approximately 0.1633, or 16.33%.