Question

A box of light bulbs has 6 75W bulbs, 4 40W bulbs, and 5 60W bulbs. If 2 bulbs are taken from the box and one is a 75W bulb, what is the probability that both are 75W? If neither bulb is 75W what is the probability that both have the same wattage?

Answer 1

This is a typical example of the urn-model. You may as well take them one by one.

Explanation:

(1)
Say the first is a 75. Then there are 5 75's left on a total of 14.
`P=5/14`

(2)
The first is not a 75, so either a 60 or a 40:

(a) first is a 60 (chance 5/15).
Then there are 4 60's left on a total of 14
Multiply because of first AND second being 60:
`5/15*4/14=20/210` (chance of 2 60'2)

(b) first is a 40 (chance 4/15).
Then there are 3 40's left on a total of 14
Multiply because of first AND second being 40:
`4/15*3/14=12/210` (chance of 2 40'2)

Now add these two, because it's EITHER 2 60's OR two 40's:
`20/120+12/120=32/120=4/15`