A box in a supply room contains 15 compact fluorescent lightbulbs, of which 5 are rated 13-watt, 6 are rated 18-watt, and 4 are rated 23-watt. Suppose that three of these bulbs are randomly selected. a. What is the probability that exactly two of the selected bulbs are rated 23-watt? b. What is the probability that all three of the bulbs have the same rating? c. What is the probability that one bulb of each type is selected? d. If bulbs are selected one by one until a 23-watt bulb is obtained, what is the probability that it is necessary to examine at least 6 bulbs?

Answer : Step 1 : Given a box in a supply room contains 15 compact fluorescent light bulbs, of which 5 are rated 13-watt, 6 are rated 18-watt, and 4 are rated 23-watt. Suppose that three of these bulbs are randomly selected. a). There is a total of 5 + 6 + 4 = 15 bulbs. Three of these bulbs are randomly selected. So15c 3 Then (4c )2(11c )1 (2 of the four 25-watt bulbs, 1 of the remaining 11 bulbs). Now we have to find the probability that exactly two of the selected bulbs are rated 23-watt. (4c ) (11c ) P (two are 25-W) = 2 1 (15c 3 (6) (11) P (two are 25-W) = (455) (66) P (two are 25-W) = (455) P (two are 25-W) = 0.145 Therefore the probability that exactly two of the selected bulbs are rated 23-watt is 0.145. Step 2 : b). Let the probability that all three of the bulbs have the same rating. Break into three cases: 3 of one type of bulb : (5c3) (6c 3 (4c 3 P(all three of the bulbs ) = (15c ) 3 (10) + (20) + (4) P(all three of the bulbs ) = (455) (34) P(all three of the bulbs ) = (455) P(all three of the bulbs ) = 0.0747