What Is the Probability That Both Peter and Simon Get Poisoned?

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The discussion revolves around calculating the probability that both Peter and Simon get poisoned after sharing a bag of fruits, with the condition that their dog ate a healthy fruit. The problem is reformulated to consider 10 fruits, with 3 poisoned, and the focus shifts to determining the joint probability of both Peter and Simon consuming poisoned fruits. Participants debate the correct approach, emphasizing the need to calculate the probability of neither getting poisoned and then using that to find the desired probability. There is confusion about whether to calculate the joint probability directly or through complementary probabilities. The conversation highlights the complexity of calculating joint probabilities in scenarios involving multiple subjects and conditions.
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Homework Statement



Peter and Simon shares a bag of 11 fruits, of which 3 are poisoned. Peter eats 4 fruits, Simon eats 3 and their dog eats 1 fruit.

What is the probability that both Peter and Simon gets poisoned, given that the dog ate a healthy fruit?

Homework Equations





The Attempt at a Solution



The dog simply removed a healthy fruit, so we can reformulate the question as: Given 10 fruits of which 3 are poisoned, what is the probability both Peter (4 fruits) and Simon (6 fruits) gets poisoned?

A = Peter gets poisoned
B = Simon gets poisoned
XcY = X choose Y

P(A) = (3c3*7c1 + 3c2*7c2 + 3c1*7c3)/10c4

P(A\capB) = (3c2*7c2*1c1*5c5 + 3c1*7c3*2c1*6c4 + 3c1*7c3*2c2*6c3)/(10c4*6c5)

P(B|A) = P(A\capB)/P(B)

This should work I believe. But I feel like I'm deriving the Fourier heat equation to figure out the boiling point of water. Is there an easier way?
 
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Either of Peter or Simon will be poisoned if they eat at least one poisoned fruit, right? So it is sufficient to calculate the probabllity they get NO poisoned fruit, then subtract from 1.
 
Gauss M.D. said:
P(B|A) = P(A\capB)/P(B)

This is not correct.
 
HallsofIvy said:
Either of Peter or Simon will be poisoned if they eat at least one poisoned fruit, right? So it is sufficient to calculate the probability they get NO poisoned fruit, then subtract from 1.
It asks for the prob that both get poisoned, not that at least one gets poisoned. If you're applying that individually it's true, but that still doesn't deal with the joint probability.
Or maybe you meant this: P(A&B) = 1 - P(!A) - P(!B) + P(!A&!B) (! signifying NOT).
 
Hint: Get a formula for P(A or B).
 

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