Probability of Permutations/Combinations

Question

How do you set up the formula for the probability of a permutation/combination?

Question: If you have a group of candy with $2$ Snickers, $4$ Kit Kats, and $2$ Butterfingers and you take two pieces out, what's the probability that those two pieces will be Kit Kats?

Answer 1

• Possible ways: out of $2+4+2=8$ choose $2$. You can do that in $\binom{8}{2}$ ways.

• Favourable ways: out of $4$ choose $2$ and out of the rest choose $0$. You can do that in $\binom{4}{2}\binom{2}{0}\binom{2}{0}$ ways.

Hence, since every way is equally likely to occur, the required probability is equal to $$\dfrac{\dbinom{4}{2}\dbinom{2}{0}\dbinom{2}{0}}{\dbinom{8}{2}}=\frac{4!2!6!}{2!2!8!}=\frac{3}{14}$$

Answer 2

I guess this is the easy way!

Probability that the first candy chosen will be a Kitkat is $\dfrac{4}{8}$

after removing one Kitkat from the group,

probability that second candy chosen will be a Kitkat is $\dfrac{3}{7}$

And Then

$$\dfrac{4}{8}\cdot\dfrac{3}{7}=\frac{3}{14}$$

Answer 3

The number of ways to take $2$ out of $2+4+2$ candies is $\dbinom{2+4+2}{2}=28$

The number of ways to take $2$ out of $4$ Kit-Kat candies is $\dbinom{4}{2}=6$

Hence the probability is $\dfrac{6}{28}$