1 In a promotion for a new type of cereal, a toy dinosaur is included in each pack. There are three different types of dinosaur to collect. They are distributed, with equal probability, randomly and independently in the packs. Sam is trying to collect all three of the dinosaurs.
- Find the probability that Sam has to open only 3 packs in order to collect all three dinosaurs.
Sam continues to open packs until she has collected all three dinosaurs, but once she has opened 6 packs she gives up even if she has not found all three. The random variable \(X\) represents the number of packs which Sam opens.
- Complete the table below, using the copy in the Printed Answer Booklet, to show the probability distribution of \(X\).
| \(r\) | 3 | 4 | 5 | 6 |
| \(\mathrm { P } ( X = r )\) | | \(\frac { 2 } { 9 }\) | \(\frac { 14 } { 81 }\) | |
- \(\mathrm { E } ( X )\) and
- \(\operatorname { Var } ( X )\).