# Thread: Probability for exactly one of two events

1. ## Probability for exactly one of two events

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.

2. As you said this is an Xor.
You want $\displaystyle P\left( {A \cap \overline B } \right) + P\left( {\overline A \cap B} \right)$, where $\displaystyle \overline B$ is not B.

Notice that $\displaystyle P\left( {A \cap \overline B } \right) = P(A) - P(A \cap B)$.

3. Originally Posted by Plato
As you said this is an Xor.
You want $\displaystyle P\left( {A \cap \overline B } \right) + P\left( {\overline A \cap B} \right)$, where $\displaystyle \overline B$ is not B.

Notice that $\displaystyle P\left( {A \cap \overline B } \right) = P(A) - P(A \cap B)$.
Thank you, so then it follows that $\displaystyle P\left( {\overline A \cap B} \right) = P(B) - P(A \cap B)$.

Which when you take $\displaystyle P\left( {A \cap \overline B } \right) + P\left( {\overline A \cap B} \right) = P(A) + P(B) - P(A \cap B) - P(A \cap B)$. But taking away the intersection twice is equivalent to taking it away once since the second subtraction would already be empty. Correct?\

I see my error, we are dealing with the probabilities so you would subtract the intersection twice!

4. $\displaystyle P\left( {A \cap \overline B } \right) + P\left( {\overline A \cap B} \right) = P(A) + P(B) - 2P(A \cap B)$

5. Yes, thanks, I was editting my reply but you drew and shot first.

6. That is the standard counting rule for the XOR.
You will use it for the symmetric difference operator.

### probability of exactly one event

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