A utility company offers a lifeline rate to any household whose electricity usage 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 $\displaystyle P(A) = .8, P(B) = .7$, and $\displaystyle P(A\cup B)=.9$. Compute the following:

a) $\displaystyle P(A\cap B)$ . $\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)$. So $\displaystyle P(A\cap B)$= .8 + .7 - .9 = .6

b) The probability that the lifeline usage amount is exceeded in exactly one of the two months. Describe this event in terms of A and B. I believe this would be an XOR relationship between A and B, so that would be $\displaystyle P(A\cup B)-P(A\cap B) = .9 - .6 = .3.$

Please let me know if these calculations and set relationships are accurate.