# Urn Model

1. May 16, 2006

### blumfeld0

an urn contains 4 red balls and 4 white balls
an experiment consists of selecting at random a sample of 4 balls and
recording the number of red balls in the sample
setup the probability distribution and compute its mean and variance

i know what a probablity distribution is. can someone please how to calculate A probability and i can calculate the rest
also i know mean = total * probability of success = what numbers exactly?

variance = SQRT(total*sucsess * failure) = what numbers exactly?

2. May 16, 2006

### arunbg

Do you know anything about binomial probability distribution ?

3. May 16, 2006

### HallsofIvy

The first thing we need to know is whether this is "sampling with replacement" or "sampling without replacement". That is, whether a ball, after it is drawn from the urn and its color recorded is or is not returned to the urn.

Obviously, there are 5 possible outcomes: 0, 1, 2, 3, or 4 red balls in the sample.

Assuming "sampling with replacement", the probability of drawing a red ball is 4/8= 1/2 and the probability of drawing a white ball is 1/2 on each draw. In order to get 0 red balls, you have to draw a white ball each time: the probability of that is (1/2)(1/2)(1/2)(1/2)= 1/16. In order to get exactly 1 red ball, you will also need to consider the different orders in which it can be done: "red, white, white, white", "white, red, white, white", etc. That's where the binomial coefficient and the binomial probability distribution arnbg mentioned comes in.

If, on the other hand, this is "sampling without replacement", it's a lot harder! The probability of getting a white ball on the first draw is still 1/2 but IF that happens, there are now 4 red and 3 white balls in the urn. The probability of getting a second white ball is now 3/7. Then there are 4 red and 2 white balls in the urn. The probability of a third white ball is 2/6= 1/3. Finally, there are 4 red and only 1 white ball in the urn. The probability of drawing a fourth white ball is 1/5. The probability of 0 red balls (4 white balls) is (1/2)(3/7)(1/3)(1/5)= 1/70.