Question

7. A jar contains 6 white beads and 3 black beads. Beads are chosen randomly from the jar one at a time until the third time a black bead turns up. a. Suppose that each bead is replaced before the next is chosen. How many beads should you expect to be chosen in the course of the experiment? [5] b. Suppose that if a bead is white, it is not replaced before the next bead is chosen, but if it is black, it is replaced before the next is chosen. How many beads should you expect to be chosen in the course of the experiment? [10] 8. For each i = 1, 2, ..., 10, X, is a random variable that gives 0 or 1 if the ith toss of a fair coin came up T or H, respectively. Let X = X1 + X2 + ... + X 10. a. Compute the expected value E(X) and variance V(X) of X. (5) b. What is the probability function of X? (10)

Answer 1

7)

Answer:

Given that:

A jar contains 6 white beads and 3 black beads. Beads are chosen randomly from the jar one at a time until the third time a black bead turns up.

Let W represents the white bead ,B represent the Block bead

Bead are chosen randomly until the third time a black beads turns up

a) Suppose that each bead is replaced before the next is chosen. How many beads should you expect to be chosen in the course of the experiment?

On this each bead it replaced before the next case it chosen

So,number of ways we can select 3 bead using the above condition is BBB+BWB+WBB+WWB

Here the number of ways of choosing a black bead is 3 and number of ways of choosing a white bead is 6

So the required number of ways is

= 3*3*3+3*6*3+6*3*3+6*6*3

= 27+54+54+108

= 243

So the number of beads should we expect to be chosen in this course of the experiment is 243

b) Suppose that if a bead is white, it is not replaced before the next bead is chosen, but if it is black, it is replaced before the next is chosen. How many beads should you expect to be chosen in the course of the experiment?

If a white bead if chosen than it is not replaced before the next bead is chosen but if it is black,it is replaced before the next is chosen

So the number of ways of choosing beads in this case is

= BBB+BWB+WBB+WWB

= 3*3*3+3*6*3+6*3*3+6*5*3

= 27+54+54+90

= 225

So the number of beads should we expect to be chosen in this course of the experiment is 225

As per HOMEWORKLIB RULES we should solve only the first question. So I have done it.
For the other question please post differently mentioning your requirement