Question

There are 7 charitable donors at a gala - 6 "typical" donors and 1 "generous" donor. Everytime a "typical" charitable donor is approached for money, she/he will donate (with equal probability) either $0, $1, $2, $4, or $8. Everytime the "generous" charitable donor is approached, she/he will always give $8. You and your friend approach the same randomly selected charitable donor and both ask for money, and you both receive $8. What is the probability that the charitable donor is the "generous" one?

Answer 1

9%

Explanation:

• The probability to get $8 in two approach and get the $8 in both cases for the generous one

The probability is equal to the probability of get $8 in the first approach times the probability of approach to the generous one plus the probability of get $8 in the second times the probability of approach to the generous one. Like this event are independent (the first approach will no change the probability of the second approach). Then we can just multiply times 2 the probability of get $8 in an approach times the probability of approach to the generous one.

P (to get $8 in two approach and get the $8 in both cases for the generous one) = 2 * P (obtain $8 in an approach) * P (to approach to charity generous)

• Lets determinate the probability of obtain $8 in a approach

This will be the probability of approach to each charity and get $8

P (obtain $8 in a approach) = P (obtain $8 in an approach to charity 1) + P (obtain $8 in an approach to charity 2) + P (obtain $8 in an approach to charity 3) + P (obtain $8 in an approach to charity 4) + P (obtain $8 in an approach to charity 5) + P (obtain $8 in an approach to charity 6) + P (obtain $8 in an approach to charity 7)

The probability of obtain $8 in an approach to charity 1,2,3,4,5 or 6 is the same, so

P (obtain $8 in a approach) = 6* P (obtain $8 in an approach to charity 1,2,3,4,5, or 6) + P (obtain $8 in an approach to charity 7)

• Probability to obtain $8 in a approach to charity 1,2,3,4,5, or 6

This will be the probability to approach to one of this charity , for the probability of obtain $8 in that approach

P (obtain $8 in an approach to charity 1,2,3,4,5, or 6)= P (to approach to charity 1,2,3,4,5, or 6) * P (obtain $8)

There are 7 charity, so the probability to approach to one of them is
`(1)/(7)`

The charity 1,2,3,4,5, or 6, will donate $0, $1, $2, $4, or $8 (5 options), so the probability to donate $8 is
`(1)/(5)`

P (obtain $8 in an approach to charity 1,2,3,4,5, or 6)= P (to approach to charity 1,2,3,4,5, or 6) * P (obtain $8) =
`(1)/(7) * (1)/(5)` =
`(1)/(35)`

• Probability to obtain $8 in a approach to charity 7

This will be the probability to approach to charity 7, for the probability of obtain $8 in that approach

P (obtain $8 in an approach to charity 7)= P (to approach to charity 7) * P (obtain $8)

There are 7 charity, so the probability to approach to one of them is
`(1)/(7)`

The charity 7 will donate $8 in each approach so the probability is 100 %

P (obtain $8 in an approach to charity 7)= P (to approach to charity 7) * P (obtain $8) =
`(1)/(7) * 1`

P (obtain $8 in a approach) = 6* P (obtain $8 in an approach to charity 1,2,3,4,5, or 6) + P (obtain $8 in an approach to charity 7)

P (obtain $8 in a approach) = 6*
`(1)/(35)` +
`(1)/(7)`

P (obtain $8 in a approach) =
`(11)/(35)`

P (to get $8 in two approach and get the $8 in both cases for the generous one) = 2 * P (obtain $8 in an approach) * P (to approach to charity generous) = 2 *
`(11)/(35)` *
`(1)/(7)` =
`(22)/(245)` = 0.089

So the probability to get $8 in two approach and get the $8 in both cases for the generous one is around 9%