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.)
a) The first two sales on Wednesday are both for more than $ 100.
b) None of the first three sales on Wednesday is for more than $ 100
To find the probabilities for both scenarios, we can use the concept of conditional probabilities.
a) The probability that the first sale is over $100 is given as 0.70. Since the sales on each day are treated as independent events, the probability of the second sale being over $100 is also 0.70. To find the probability of both events occurring, we multiply the individual probabilities:
P(First two sales over $100) = P(First sale over $100) * P(Second sale over $100) = 0.70 * 0.70 = 0.49
Therefore, the probability that the first two sales on Wednesday are both for more than $100 is 0.49.
b) The probability that the first sale is not over $100 is the complement of the probability that it is over $100, which is 1 - 0.70 = 0.30. Again, since each sale is treated as an independent event, the probability of the second and third sales not being over $100 is also 0.30 each. To find the probability of none of the first three sales being over $100, we multiply the individual probabilities:
P(None of the first three sales over $100) = P(First sale not over $100) * P(Second sale not over $100) * P(Third sale not over $100)
= 0.30 * 0.30 * 0.30 = 0.027
Therefore, the probability that none of the first three sales on Wednesday is for more than $100 is 0.027.