What's the probability of a household subscribing to at least one newspaper?

[caljuice]

Homework Statement

If 60% of households subscribe to Metro(M) newspaper, 80% subscribe to local (L) newspaper, and 50% subscribe to both,

1)what's the probability that a random household subscribes to at least one paper?
2) what's the probability that a random household subscribes to exactly one paper?

The Attempt at a Solution

1) The probability of at least one paper subscribed is P(M U L) = P(M) + P(L) - P(M AND L) which gives the answer of 0.9. But I'm wondering why this works? The phrasing "at least" means the possibilities can be just P(M), just P(L), or P(M AND L) but from the equation we are substracting out P(M AND L), which means P(M) and P(L) are the only possible outcomes. What's wrong with my thinking here?

2) Probability = P(M^c and L) U P(M^c and L) and they are mutually exclusive so you just sum the two probabilities.

P^c = 1- P
Not sure how to find P(M^c and L) since they aren't independent.

thanks.

 

[Ray Vickson]

Homework Statement

If 60% of households subscribe to Metro(M) newspaper, 80% subscribe to local (L) newspaper, and 50% subscribe to both,

1)what's the probability that a random household subscribes to at least one paper?
2) what's the probability that a random household subscribes to exactly one paper?

The Attempt at a Solution

1) The probability of at least one paper subscribed is P(M U L) = P(M) + P(L) - P(M AND L) which gives the answer of 0.9. But I'm wondering why this works? The phrasing "at least" means the possibilities can be just P(M), just P(L), or P(M AND L) but from the equation we are substracting out P(M AND L), which means P(M) and P(L) are the only possible outcomes. What's wrong with my thinking here?

2) Probability = P(M^c and L) U P(M^c and L) and they are mutually exclusive so you just sum the two probabilities.

P^c = 1- P
Not sure how to find P(M^c and L) since they aren't independent.

thanks.

P(L & M) is part of P(M) and is also part of P(L), so the summation P(L) + P(M) counts P(L&M) twice. That is why you need to subtract it---so you only count it once.

RGV

 

[HallsofIvy]

Imagine that there are 100 households. Then 60 subscribe to M and 50 subscribe to both so 10 subscribe to M only. 80 subscribe to L and 50 subscribe to both so 30 subscribe to L only. That makes a total of 10+ 50+ 30= 90 households out of 100 that subscribe to at least one newspaper and 10+ 30= 40 that subscribe to exactly one newspaper.