Question

suppose an urn contains r red, w white, and b black balls. let x1, . . . , xk denote k draws from the urn with replacement, and y1, . . . , yk denote k draws from the urn without replacement. what is the relation between h(x1, . . . , xk) and h(y1, . . . , yk)? give a formal argument.

Answer 1

The relation between h(x1, . . . , xk) and h(y1, . . . , yk) can be explained as follows:

1. With Replacement (h(x1, . . . , xk)):

When draws are made from the urn with replacement, each draw is independent of the others. This means that after each draw, the ball is put back into the urn before the next draw is made. The probability of drawing a specific ball remains the same for each draw, regardless of previous outcomes. Therefore, h(x1, . . . , xk) represents a sequence of independent events.

2. Without Replacement (h(y1, . . . , yk)):

When draws are made from the urn without replacement, each draw affects the probability of the next draw. As each ball is drawn, the total number of balls in the urn decreases, influencing the probabilities of subsequent draws. The likelihood of drawing a specific ball changes with each draw, as the composition of the urn changes. Therefore, h(y1, . . . , yk) represents a sequence of dependent events.

In summary, the relation between h(x1, . . . , xk) and h(y1, . . . , yk) is that h(x1, . . . , xk) represents independent events where each draw is unaffected by previous outcomes, while h(y1, . . . , yk) represents dependent events where each draw affects the probability of subsequent draws due to the absence of replacement.