Question

Suppose you have a bag with 6 blue marbles and 4 red marbles. Calculate the following probabilities:

a) If you randomly remove half of the marbles from the bag, what is the probability that 3 of them are blue and 2 are red?

b) Suppose that you sample with replacement from the bag: you take out a marble, record its color, then throw it back in. What is the probability that you need to repeat this process exactly 10 times before seeing your first red marble on your 10th selection?

c) Again sampling with replacement, what is the probability that you need to repeat the process exactly 10 times before seeing your 4 th red marble on your 10th selection?

d) Again sampling with replacement, what is the probability that if you sample 10 times, exactly 4 of your marbles are red?

Answer 1

Total number of marbles = 6+4 = 10

a)

Number of ways of selecting 5 marbles out of 10 is C(10,5).

Number of ways of selecting 3 blue marbles out of 6 and 2 red marbles out of 4 is

C(6,3) * C(4,2)

The probability that 3 of them are blue and 2 are red is

b)

Since draws are with replacement so in each draw probability of getting a blue marble is

P(blue) = 6/10 = 0.60

P(red) = 4/10 =0.40

Using geometric distribution the probability that you need to repeat this process exactly 10 times before seeing your first red marble on your 10th selection is

c)

Here we need to use negative binomial distribution with parameters as follow:

n= 10, r = 4

d)

Here we need to use binomial distribution with following parameters:

n=10 and p=0.4

The probability that if you sample 10 times, exactly 4 of your marbles are red is