Question 12.10E4: DVD Selection Jennifer Alfredo has 4 Scooby Doo DVDs, 5 Magi......

a)  If no Scooby Doo DVDs are to be selected, then only Magic School Bus and Bob the Builder DVDs must be selected. Eight DVDs are either Magic School Bus or Bob the Builder. Thus, the number of ways that 4 Magic School Bus or Bob the Builder DVDs may be selected from the possible 8 Magic School Bus or Bob the Builder DVDs is _8C_4. The total number of possible selections is _{12}C_4.

\text{P(no Scooby Doo DVDs) = }\frac{_8C_4}{_{12}C_4}  =  \frac{70}{495}  =  \frac{14}{99}

b)  When 4 DVDs are selected, the choice must contain either no Scooby Doo DVDs or at least 1 Scooby Doo DVD. Since one of these outcomes must occur, the sum of the probabilities must be 1, or

P(no Scooby Doo DVDs) + P(at least 1 Scooby Doo DVD) = 1

Therefore,

P (at least 1 Scooby Doo DVD) = 1- P (no Scooby Doo DVDs)
=  1  –  \frac{14}{99}  =  \frac{99}{99}  –  \frac{14}{99}  =  \frac{85}{99}

Note that the probability of selecting no ScoobyDoo DVDs, \frac{14}{99}, was found in part (a).

c)  The number of ways of selecting 2 Scooby Doo DVDs out of 4 Scooby Doo DVDs is _4C_2, which equals 6. The number of ways of selecting 2 Magic School Bus DVDs out of 5 Magic School Bus DVDs is _5C_2, which equals 10. The total number of possible selections when 4 DVDs are selected from the 12 choices is _{12}C_4, which equals 495. Since both the 2 Scooby Doo and 2 Magic School Bus DVDs must be selected, the probability is calculated as follows.