1988 A-Level H2 Mathematics

In a sales campaign, a petrol company gives each motorist who buys their petrol a card with a picture of a film star on it. There are ten different pictures, one each of ten different film stars, and any motorist who collects a complete set of all ten pictures gets a free gift. On any occasion when a motorist buys petrol, the card received is equally likely to carry any one of the ten pictures in the set.

1. the first four cards of the motorist receives all carry different pictures.[3]
2. the first four cards received result in the motorist having exactly three different pictures.[3]
3. Two of the ten film stars in the set are \(X\) and \(Y\). Find the probability that the first four cards received result in the motorist having a picture of \(X\) or \(Y\) (or both).[4]
   [Note: a picture here is interpreted as at least one picture and not exactly one picture]
4. At a certain stage the motorist has collected nine of the ten pictures. Find the least value of \(n\) such that probability that he needs at most \(n\) more cards to complete the set is greater than \(0.99\).[4]