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?

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?

Grace Agate · Accepted Answer

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# 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).

Anthony Arnold · Answer

undefined To find the likelihood that two people go to the market on a given morning, we need to consider the probabilities of different combinations of two people going to the market.

Person 1 and Person 2:
Probability = (1/2) * (2/3) = 1/3

Person 1 and Person 3:
Probability = (1/2) * (3/4) = 3/8

Person 2 and Person 3:
Probability = (2/3) * (3/4) = 1/2

Adding these probabilities together:
1/3 + 3/8 + 1/2 = 8/24 + 9/24 + 12/24 = 29/24

So, the likelihood that two people go to the market on a given morning is 29/24.