High School

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A utility company offers a lifeline rate to any household whose electricity falls below 240 kWh during a particular month. Let \( A \) denote the event that a randomly selected household in a certain community does not exceed the lifeline usage during January, and let \( B \) be the analogous event for the month of July (A and B refer to the same household). Suppose \( P(A) = 0.8 \), \( P(B) = 0.7 \), and \( P(A \cup B) = 0.9 \).

Compute the following:

A. \( P(A \cap B) \)

B. The probability that the lifeline usage amount was exceeded in exactly one of the two months. Describe this event in terms of \( A \) and \( B \).

Answer :

Answer:

A. 0.6

B. 0.3

Step-by-step explanation:

For part A we use the the general probability addition rule for the union of two events that states

P(A∪B) = P(A) + P(B) − P(A∩B)

Making P(A∩B) the subject of the equation above

P(A∩B) = P(A) + P(B) − P(A∪B)

P(A∩B) = 0.8 + 0.7 - 0.9

P(A∩B) = 0.6

B.

The description in terms of A and B is:

P(A but not B) + P(B but not A) = P(A∩B') + P(B∩A')

where A' is compliment of set A and B' is compliment of set B

The above description means either it exceeds in July or in January (exactly in one of the two months)

P(A∩B') = 0.8 − 0.6

P(B∩A') = 0.7 − 0.6

P(A∩B') + P(B∩A') = 0.3

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