. A box in a certain supply room contains four 40-W lightbulbs, five 60-W bulbs, and six 75-W bulbs. Suppose that three bulbs are randomly selected. 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?

Answer:a) 59.34%b) 44.82%c) 26.37%d) 4.19%Step-by-step explanation:(a) There are in total 4+5+6 = 15 bulbs. If we want to select 3 randomly there are  K ways of doing this, where K is the combination of 15 elements taken 3 at a time [tex]K=\binom{15}{3}=\frac{15!}{3!(15-3)!}=\frac{15!}{3!12!}=\frac{15.14.13}{6}=455[/tex] As there are 9 non 75-W bulbs, by the fundamental rule of counting, there are 6*5*9 = 270 ways of selecting 3 bulbs with exactly two 75-W bulbs. So, the probability of selecting exactly 2 bulbs of 75 W is [tex]\frac{270}{455}=0.5934=59.34\%[/tex] (b) The probability of selecting three 40-W bulbs is [tex]\frac{4*3*2}{455}=0.0527=5.27\%[/tex] The probability of selecting three 60-W bulbs is [tex]\frac{5*4*3}{455}=0.1318=13.18\%[/tex] The probability of selecting three 75-W bulbs is [tex]\frac{6*5*4}{455}=0.2637=26.37\%[/tex] Since the events are disjoint, the probability of taking 3 bulbs of the same kind is the sum 0.0527+0.1318+0.2637 = 0.4482 = 44.82% (c) There are 6*5*4 ways of selecting one bulb of each type, so the probability of selecting 3 bulbs of each type is [tex]\frac{6*5*4}{455}=0.2637=26.37\%[/tex] (d) The probability that it is necessary to examine at least six bulbs until a 75-W bulb is found, supposing there is no replacement, is the same as the probability of taking 5 bulbs one after another without replacement and none of them is 75-W. As there are 15 bulbs and 9 of them are not 75-W, the probability a non 75-W bulb is [tex]\frac{9}{15}=0.6[/tex] Since there are no replacement, the probability of taking a second non 75-W bulb is now [tex]\frac{8}{14}=0.5714[/tex] Following this procedure 5 times, we find the probabilities [tex]\frac{9}{15},\frac{8}{14},\frac{7}{13},\frac{6}{12},\frac{5}{11}[/tex] which are 0.6, 0.5714, 0.5384, 0.5, 0.4545 As the events are independent, the probability of choosing 5 non 75-W bulbs is the product 0.6*0.5714*0.5384*0.5*0.4545 = 0.0419 = 4.19%