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?

1 Answer
MeneerNask
Nov 6, 2015

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).

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