# 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?

Sep 27, 2017

$P r = \frac{91}{285}$.

#### 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
$P r \left(a\right) = \frac{7}{10}$ and $P r \left(b\right) = \frac{3}{10}$.

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 $\left(\begin{matrix}14 \\ 3\end{matrix}\right)$ by the total number of possible groups of 3 $\left(\begin{matrix}20 \\ 3\end{matrix}\right)$. 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!))

$= \frac{14 \times 13 \times 12}{20 \times 19 \times 18}$

$= \frac{7 \times 13 \times 1}{5 \times 19 \times 3} \text{ "=" } \frac{91}{285}$

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 $P r \left({B}_{1}\right) = \frac{7}{10.}$ 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 $P r \left({B}_{2} | {B}_{1}\right) = \frac{13}{19}$.

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 $P r \left({B}_{3} | {B}_{1} , {B}_{2}\right) = \frac{12}{18} ,$ or $\frac{2}{3}$.

To find the final probability, we multiply these three fractions together:
$\frac{7}{10} \cdot \frac{13}{19} \cdot \frac{2}{3}$
$= \frac{182}{570}$
$= \frac{91}{285}$ 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!