# probability specific people appear in a sample from a group

The Justice League wants to randomly select a group of 7 from among the 40 currently available superheroes/superheroines to investigate a glowing meteorite. What is the probability that Bat Man, Wonder Woman, and Spider Man are all chosen?

Initially, I tried to simplify this problem as follows: Bob, Susan, and Mary are standing near the meteorite. What's the probability that both Bob and Susan are randomly zapped with a mutagenic beam?

I drew out the probability space for this sub-problem and get $P(s,b\ zapped)= \frac{2}{9}$ However, this is with replacement (e.g., Bob can get zapped twice).

   b  s  m
b  .  z  .
s  z  .  .
s  .  .  .


When it is without replacement, I am not sure. Is it $P(s,b\ zapped)=\frac {1}{3}\frac{1}{2}$ or do I need to take into account the different ways people can get zapped, $P(s,b\ zapped)={3\choose 2}\frac {1}{3}\frac{1}{2}$ ?

For the original problem, my initial guess was to try to account for the different ways all three are chosen from seven: $P(B,S,W chosen)= {7 \choose 3}{1 \over 40} {1 \over 39} {1 \over 38} {37 \over 37}{36 \over 36}{ 35 \over 35}{ 34 \over 34}$

However if all three are chosen last then it would seem to be: $P(B,S,W chosen\ last)= {7 \choose 3}{40 \over 40} {39 \over 39} {38 \over 38} {37 \over 37}{1 \over 36}{ 1 \over 35}{ 1 \over 34}$

I also tried to think of treating Bat Man, Wonder Woman, and Spider man as a generic group of 3, but I still seem to be missing the possibility of having those three selected in different orderings, like second, fourth, and last.

$P(B,S,W chosen)= {3 \over 40} {2 \over 39} {1 \over 38} {37 \over 37}{36 \over 36}{ 35 \over 35}{ 34 \over 34}$

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## 1 Answer

The probability of choosing a "good" team is the ratio of "good" cases (where all three named superheroes are chosen) and all possibles ones.

(All): How many ways can you choose 7 people out of 40 (the order in which those people are chosen does not matter)?

(Good): If three of the seven people are clear from the beginning, your only choice is to find four others who will complete the team. How many ways can you choose those four from the remaining superheroes?

Hint: Binomial coefficients provide the answers to both parts.

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Ah, it was a hypergeometric in disguise. I should've known. –  maogenc May 18 '13 at 19:47