N2019 P2 Q7

A company produces drinking mugs. It is known that, on average, \(8\%\) of the mugs are faulty. Each day the quality manager collects \(50\) of the mugs at random and checks them; the number of faulty mugs found is the random variable \(F\).

1. State, in the context of the question, two assumptions needed to model \(F\) by a binomial distribution.[2]

You are now given that \(F\) can be modelled by a binomial distribution.

2. Find the probability that, on a randomly chosen day, at least \(7\) faulty mugs are found.[2]
3. The number of faulty mugs produced each day is independent of other days. Find the probability that, in a randomly chosen working week of \(5\) days, at least \(7\) faulty mugs are found on no more than \(2\) days.[2]

The company also makes saucers. The number of faulty saucers also follows a binomial distribution. The probability that a saucer is faulty is \(p\). Faults on saucers are independent of faults on mugs.

4. Write down an expression in terms of \(p\) for the probability that, in a random sample of \(10\) saucers, exactly \(2\) are faulty.[1]

The mugs and saucers are sold in sets of \(2\) randomly chosen mugs and \(2\) randomly chosen saucers. The probability that a set contains at most \(1\) faulty item is \(0.97\).

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