Question

Three people go to the market in the morning with the following probabilities: 1/2, 2/3, and 3/4. What is the likelihood that two go to the market on a given morning?

Answer 1

We'll have to work out the probabilities for all three combinations of two, where one stays home . Let's call the people `A,B,C`

Explanation:

We'll define the probabilities by the letter `P`
(the `not` sign means "not", `^^` means "and")
`P(A)=1/2->P(notA)=1/2`
`P(B)=2/3->P(notB)=1/3`
`P(C)=3/4->P(notC)=1/4`

Then we get (remember "and" means multiply)
`P(A^^B^^notC)=1/2*2/3*1/4=2/24`
`P(A^^notB^^C)=1/2*1/3*3/4=3/24`
`P(notA^^B^^C)=1/2*2/3*3/4=6/24`

Since these combinations are mutually exclusive we add :
`P("total")=2/24+3/24+6/24=11/24`

Note :
If the question had been "at least two" we would have to add:
`P(A^^B^^C)=1/2*2/3*3/4=6/24` for a total of `17/24`

If you had just added the probabilities of
`P(A^^B)+P(A^^C)+P(B^^C)` without the NOT's, then `P(A^^B^^C)` would have been included three times. In fact `P("total")` would then be greater than 1 (a capital sin).