A box in a certain supply room contains four 40-W lightbulbs, three 60-W bulbs,

Question

A box in a certain supply room contains four 40-W lightbulbs, three 60-W bulbs, and five 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?

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

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

(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?

Explanation / Answer

There are (12 C 3) = 220 ways to choose 3 bulbs.

a) Exactly two 75W can happen (5C2) = 10 ways.
One non-75W can happen (7C1) = 7 ways.
Combination can happen 10 * 7 = 170 ways.
Probability = 170/220 = .77 or 77%

b) All three the same can happen 3 ways; 40-40-40, 60-60-60, 75-75-75.
(4C3) + (3C3) + (5C3) = 4 + 1 + 10 = 15 ways
Probability = 15 /220 = .07678 or 7.678%

c) One of each can happen (4C1) * (4C1) * (7C1) = 112 ways
Probability = 112 / 560 = .0681 or 6.81%

d) The only way that at least 6 bulbs are required is if the first 5 are chosen from the 7 non-75W bulbs.

There are (12 C 5) = 792 ways to choose the first 5 bulbs.
Of those (7C5) = 21 ways to choose from only the non-75W bulbs
Probability = 21 / 792 = .0265 or 2.65%

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