0
$\begingroup$

The probability that a randomly chosen person eats fruits is $0.22$, and the probability that he's healthy is $0.4$. The probability that he eats fruits AND is healthy is $0.12$. Suppose you choose $3$ people at random. What is the probability that all three people eat fruits?

Clearly, those are two mutually inclusive events, which are dependent. Here's what I did, which I believe is wrong.

$0.22^3=0.011$

This seems way too simple, and I don't know what I'm missing.

  • 0
    "The probability that a randomly chosen person eats fruit is 0.22 ... Suppose you choose 3 people at random. What is the probability that all three eat fruit?" Your answer seems right, but the wording is very important. If instead it asked "What is the probability that at least 1 of the 3 eat fruit?" then the answer would be different. – 2017-01-31
  • 0
    @joeb I see. Very helpful. So if they asked for the probability that at least 1 of the 3 eats fruits, the answer would be $0.22+0.22^2+0.22^3$. Would that be correct? – 2017-01-31
  • 1
    No. In that case it is easier to subtract from $1$ the probability that none of them eats fruit resulting in whatever $1 - (1-0.22)^3$ turns out to be. But again, if the wording of the original problem is "all three people" then you were right to begin with. I was only suggesting you make sure that was indeed what they are asking. – 2017-01-31
  • 0
    @joeb Alright, my calculations was way off. $0.22+0.22^2+0.22^3$ equals to $0.279$. Whereas $1 - (1-0.22)^3$ equals to $0.525$. I wonder why they are not logically equivalent. – 2017-01-31
  • 1
    Because $0.22$ is the probability that one person eats fruits. What about the other two? Which are the other two?$$1-(1-0.22)^3 = \binom 31~0.22~(1-0.22)^2+ \binom 32 ~0.22^2~(1-0.22)+0.22^3$$ – 2017-01-31

2 Answers 2

1

Does order matter in this case? case 1: Eats Fruit = Yes case 2: Eats Fruit = No

Combinations (order doesn't matter): Pick 1: Yes/Yes/Yes = 1/4 Pick 2: Yes/Yes/No = 1/4 Pick 3: Yes/No/No = 1/4 Pick 4: No/No/No = 1/4

Combinations (order DOES matter): 3 Yes: Yes/Yes/Yes = 0.22*0.22*0.22 2 Yes: Yes/Yes/No = 0.22*0.22*(1-0.22) 2 Yes: Yes/No/Yes = 0.22*(1-0.22)*0.22 2 Yes: No/Yes/Yes = (1-0.22)*0.22*0.22 2 No: No/No/Yes = (1-0.22)*(1-0.22)*0.22 2 No: No/Yes/No = (1-0.22)0.22(1-0.22) 2 No: Yes/No/No = 0.22*(1-0.22)(1-0.22) 3 No: No/No/No = (1-0.22)(1-0.22)*(1-0.22)

The 8 permutations must = 1

1

This seems way too simple, and I don't know what I'm missing.

You are missing nothing.   It is that simple.   The question just contains a lot of distracting details; the main task was finding what you needed and ignoring the rest.