An urn contains n white balls andm black balls. (m and n are both positive numbers.) (a) if two balls are drawn without replacement, what is the probability that both balls are the same color? (b) if two balls are drawn with replacement (i. e., one ball is drawn and it’s color recorded and then put back. then the second ball is drawn.) what is the probability that both balls are the same color. (c) show that the probability in part (b) is always larger than the one in part (a).

DISCLAIMER: Please let me rename b and w the number of black and white balls, for the sake of readability. You can switch the variable names at any time and the ideas won't change a bit!

Case 1: both balls are white.

At the beginning we have balls. We want to pick a white one, so we have a probability of of picking a white one.

If this happens, we're left with white balls and still black balls, for a total of balls. So, now, the probability of picking a white ball is

The probability of the two events happening one after the other is the product of the probabilities, so you pick two whites with probability

Case 2: both balls are black

The exact same logic leads to a probability of

These two events are mutually exclusive (we either pick two whites or two blacks!), so the total probability of picking two balls of the same colour is

Case 1: both balls are white.

In this case, nothing changes between the two picks. So, you have a probability of of picking a white ball with the first pick, and the same probability of picking a white ball with the second pick. Similarly, you have a probability of picking a black ball with both picks.

This leads to an overall probability of

Of picking two balls of the same colour.

We want to prove that

Expading all squares and products, this translates to

As you can see, this inequality comes in the form

With x and y greater than k. This inequality is true whenever the numerator is smaller than the denominator:

And this is our case, because in our case we have

step-by-step explanation: