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 $F$.

(i) 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.

(ii) Find the probability that, on a randomly chosen day, at least 7 faulty mugs are found.

[2]

(iii) 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.

(iv) 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 $0.97$.

(v) Write down an equation satisfied by $p$. Hence find the value of $p$.

[4]