Probability question?

Question

A strain of peas has 3 green peas and 1 yellow in every four peas. If 12 peas are randomly selected, what is the probablity that exactly 8 of the peas are green?

I know that the green would have .75 probability and yellow would have .25. But I'm not sure where to go from here..would I have to take .75 to the 8th power or something? I need help! :(

Answer 1

This is a binomial distribution. P(k) = C(n,k) p^k (1 - p)^(n-k). In this case, n =12, the total number of experiments. k =8, the number of desired events. C(n,k) = n choose k. p = .75, the probabilty of getting a green pea.

Using Excel, we get P(8) = 0.193577707

Answer 2

Whats going on here is a little subtle. It is easiest to start by looking at pulling out 2 peas. The possbile ocutomes are:
bb,by,yb,yy.

Now you are correct that for bb the answer would be (.75)^2
and for yy the answer would be (.25)^2

Now for "by" the answer would be (.75)(.25) , howver "by" is the probalbility of choosing a black first and a yellow second.

Since for this problem we don't care about the order the peas were selected in "by" and "yb" count the same

so the probablity of picking a black and yellow so we add p(yb) and p(by)

For this question if we were going to pick 8 blacks followed by 4 yellows then the answer would be (0.75)^8 x (0,25)^4

However there are many different orders that would give us 8 black and 4 yellow. We have to count those combinations

We use C(12,8) as the count of combinations {12 choose 8} and the answer is C(12,8) x (0.75)^8 x (0,25)^4

Do you know what a factorial is? 4 Factorial, written 4 ! is cacluated by taking 4 x 3 x 2 x1 which is 24
5 ! = 5 x 4 x 3 x 2 x 1

The Combinations are writen using factorials C(n,k) =
n !
-----------
k! (n-k!)

Answer 3

Sometime this question is asked at a point in the class where you have to work it out from first principles, later there is an equation to just plug in the numbers.

you have a binomial distribution, p^4(1-p)^8

Answer 4

You need to use the binomal distribution here:

http://mathworld.wolfram.com/BinomialDistribution.html

The probability of getting exactly 8 green in 12 is:

P(8|12) = (12 choose 8)*[(0.75)^8]*[(0.25)^4]

(12 choose 8) = 12!/8!/(12-8)! = 495

P(8|12) = 495*[(0.75)^8]*[(0.25)^4] = 0.1936