2025 SAJC P2 Q8

A company sells toys in blind boxes. Each blind box contains exactly one toy. On average, \(2\%\) of such blind boxes contain a faulty toy.

A collector buys \(15\) such blind boxes at random.

1. State, in context, two assumptions needed for the number of faulty toys he gets to be well modelled by a binomial distribution.[2]

Assume now that the number of faulty toys the collector gets has a binomial distribution.

2. Find the probability that the collector gets no less than \(3\) faulty toys.[1]
3. Find the probability that the fourteenth box is the second box with a faulty toy.[2]

The company also sells keychains. The probability of a keychain being faulty is \(p\). The number of faulty keychains follows a binomial distribution. Faults on keychains are independent of faults on toys in blind boxes.

4. Write down an expression, in terms of \(p\), for the probability that in a random sample of \(3\) keychains, exactly one is faulty.[1]

A surprise pack contains \(2\) randomly chosen blind boxes and \(3\) randomly chosen keychains.

Given that a randomly selected surprise pack contains at most \(1\) faulty item, the probability that one of the toys is the only faulty item in the pack is \(0.035\).

5. Write down an equation satisfied by \(p\). Hence, find the value of \(p\).[4]