A box in a certain supply room contains four 40W lightbulbs

A box in a certain supply room contains four 40-W lightbulbs, seven 60-W bulbs, and three 75-W bulbs. Suppose that three bulbs are randomly selected. (Round your answers to four decimal places.)

(a) What is the probability that exactly two of the selected bulbs are rated 75-W?
1

(b) What is the probability that all three of the selected bulbs have the same rating?
2

(c) What is the probability that one bulb of each type is selected?
3

(d) Suppose now that bulbs are to be selected one by one until a 75-W bulb is found. What is the probability that it is necessary to examine at least six bulbs?
4

Solution

There are (16 C 3) = 560 ways to choose 3 bulbs.

a) Exactly two 75W can happen (7C2) = 21 ways.
One non-75W can happen (8C1) = 8 ways.
Combination can happen 21 * 8 = 168 ways.
Probability = 168/560 = .3 or 30%

b) All three the same can happen 3 ways; 40-40-40, 60-60-60, 75-75-75.
(4C3) + (4C3) + (7C3) = 4 + 4 + 35 = 43 ways
Probability = 43 /560 = .07678 or 7.678%

c) One of each can happen (4C1) * (4C1) * (7C1) = 112 ways
Probability = 112 / 560 = .2178 or 21.78%

d) The only way that at least 6 exams are required is if the first 5 are chosen from the 8 non-75W bulbs.

There are (16 C 5) = 4368 ways to choose the first 5 bulbs.
Of those (8C5) = 56 ways to choose from only the non-75W bulbs
Probability = 56 / 4368 = .0128 or 1.28%