binomial distribution

Question

I have a bag with 100 balls, 30 green balls and 70 red ones. If I take a ball out of the bag 20 times (with replacement), what is the probability of getting 18 green balls given by binomial distribution.

Answer 1

The key thing here is that it is sampling with replacement. That means the samples are independent, so we can use the binomial distribution.

P(green) = 30/(30+70) = 30/100 = 3/10

Then, from the binomial formula

P(18 greens in a sample of 20) = C(20,18)* (3/10)¹⁸ * (1-3/10)^20-18 =

C(20,18)* (3/10)¹⁸ * (7/10)² =

190 * 3.87420489 * 10^-10 * 49/100 =

3.60688475259 * 10^-8

Excel and statistics calculators also allow this to be calculated via a built-in function.

In Excel, binom.dist(18, 20, 3/10, 0) = 3.60688475259 * 10^-8

(The final 0 means that this is P(X = 18), not P(X <=18)

Answer 2

To get this answer you are looking for the probability of the term g^18r^2 in the binomial expansion of

(g+r)^20 where g is probability of picking green which is 30/100 or .3 and r is probability of picking red which is 70/100 or.7

To do this you also will need the coefficient of this term which is 20 choose 18 which is (20)(19)/18which equals 190/9

So the probability is(190/9)( .3^18)(.7^2)

I will let you enter into your calculator to get the result which is a very small number