Question

An urn contains 1 black marble, 1 white marble, and 1 red marble. Suppose you draw a marble from the urn 8 times with replacement.

a) Describe the sample space. How would you denote outcomes? How do you assign probabilities to each outcome?

b) What is the probability of drawing exactly 2 black and 3 white marbles after the 8 draws?

c) How many different configurations of numbers of black, white, and red marbles are possible for the 8 draws?

Answer 1

a) The sample space consists of all possible combinations of red , white and black marbles in 8 draws with replacement.

Let X, Y and Z denote the number Black , White and Red marbles drawn in each draw respectively . Let B, W and R denote Black , White and Red marbles respectively.

Each element of the sample space would be B^XW^YR^Z , that is X number of Black , Y number of W and Z number of R .

The sample space is given by

S= { B^XW^YR^Z : 0 X,Y,Z 8; X+Y+Z=8 }

Therefore each outcome is denoted by B^XW^YR^Z

As the marbles are drawn with replacement , each color has equal probability of being selected in each of the 8 draws .There are three colors , thus each color has probability =1/3

Thus Probability of each outcome is given by

P(B^XW^YR^Z)

Probability of each outcome is given by = 0.0002

b) As we have calculated that each outcome has same probability

Therefore, probability of drawing exactly 2 black and 3 white marbles in 8 draws = 0.0002

c) There are 8 draws . In each draw 1 marble is chosen out of 3 in 3 ways . Thus in 8 draws , marbles are chosen in 3⁸  ways

Number of different configurations = 3⁸