Question

30% of the 20 people in the Math Club have blonde hair. If 3 people are selected at random from the club, what is the probability that none have blonde hair?

Answer 1

`Pr=91/285`.

Explanation:

`30%` of 20 people is 6 people, so 6 people have blonde hair. This means that 14 people don't have blonde hair. If we let `a` equal non-blonde hair, and `b` equal blonde hair, that means
`Pr(a)=7/10` and `Pr(b)=3/10`.

If we think about it as picking the 3 people simultaneously, then the probability we seek is the chance of picking a group of 3 non-blondes. This is found by dividing the number of groups with 3 non-blondes `((14),(3))` by the total number of possible groups of 3 `((20),(3))`. With some simplification, we get

`Pr("3 non-blondes")=(14!)/(11!" "3!)-:(20!)/(17!" "3!)`

`=(14xx13xx12xxcancel(11!))/(cancel(11!)" "cancel(3!))-:(20xx19xx18xxcancel(17!))/(cancel(17!)" "cancel(3!))`

`=(14xx13xx12)/(20xx19xx18)`

`=(7xx13xx1)/(5xx19xx3)" "=" "91/285`

If we think about it as picking the people one by one without replacing them, then the probability of a non-blonde getting picked the first time is `Pr(B_1)=7/10.` After a successful first pick, the probability for a non-blonde getting picked the second time is the number of non-blondes left (13) divided by the number of people left (19), which gives us `Pr(B_2|B_1)=13/19`.

Finally, if we are successful on both the first and second picks, the probability of picking a non-blonde for a third time is, again, the number of non-blondes left (12) divided by the number of people left (18), which gives us `Pr(B_3|B_1, B_2)=12/18,` or `2/3`.

To find the final probability, we multiply these three fractions together:
`7/10*13/19*2/3`
`=182/570`
`=91/285` chance of picking 3 non-blondes one-by-one.

https://math.stackexchange.com/questions/941150/what-is-the-difference-between-independent-and-mutually-exclusive-events

I hope I helped!