What is the probability of a red M&M

A bag contains 50 M&Ms; 10 of them are red.

(a) If half of the M&Ms are eaten, what is the probability that 5 red ones were among them?

My work:

P(5 red) = (((10 choose 5)(40 choose 15))/(50 choose 25) = 0.08

(b) If half of the M&Ms are eaten, Find the probability that 5 red ones are among the remaining M&Ms

My work:

P(red remaining) = ((10 choose 5)(15 choose 5))/(25 choose 10) = 0.23

(c) If half of the M&Ms are eaten, how many red M&Ms can you expect to be among them?

This part I had no idea how to do.



I really wasn't sure how to do these. Are these right? Can someone explain the three parts to me?

Thank you!

(a) You should have

P(5 red) = (10 C 5)*(40 C 20) / (50 C 25) = 0.2748 to 4 significant figures.

(b) Now if 5 red are eaten, then 5 red remain. So the answer to (b)
is actually the same as to (a).

(c) For this one you need to look at each possible scenario and then
use what is called a weighted average:

(i) 0 red ones eaten -----> P(0) = (10 C 0)*(40 C 25) / (50 C 25)
(ii) 1 red one eaten -----> P(1) = (10 C 1)*(40 C 24) / (50 C 25) .....

and in general for 0 <= n <= 10 we have that for

n red ones eaten -----> P(n) = (10 C n)*(40 C (25 - n)) / (50 C 25).

Here is a table for P(n):

http://www.wolframalpha.com/input/?i=%28…

Now we need to "weight" these probabilities to get the expected number
of red ones eaten. This involves multiplying the probability for n red
ones eaten by n. This gives an expected value of

sum(n=0 to 10)(n*P(n)) = 5, after using WolframAlpha:

http://www.wolframalpha.com/input/?i=sum…

This makes intuitive sense; if we eat half the M&M's we would
"expect" to eat half of any subset of M&M's. Since the subset
of red M&M's has 10 elements we would thus expect to end up
eating 5 of them on average.

Generally speaking, if you eat half of the bag then you could probably eat half of the red ones.

Actually, 10 red M&Ms in a bag of 50 is an extremely random situation. There is a good chance that you will eat all the red ones and a good chance that you won't eat any red ones.

So your question of what are the chances of eating 5 red ones?

50 / 50

You will or you won't.

Fifty - Fifty.