2018 PJC BT2 P2 Q8
Tomatoes are sold in boxes of $16$. On average, $15\%$ of them are damaged.
(i)
State, in context, two assumptions needed for the number of damaged tomatoes in a box to be well modelled by a binomial distribution.
[2]
Assume now that the number of damaged tomatoes in a box has a binomial distribution.
(ii)
Find the most likely number of damaged tomatoes in a randomly chosen box.
[2]
(iii)
Find the probability that a box contains less than $6$ damaged tomatoes.
[1]
(iv)
A box contains less than $6$ damaged tomatoes. Find the probability that it contains at least $2$ damaged tomatoes.
[3]
(v)
Two boxes of tomatoes are randomly selected. Find the probability that one box has at least $2$ damaged tomatoes and the other box has none.
[2]
(vi)
A market trader buys $80$ boxes of tomatoes. Find the probability that there are more than $60$ boxes with at least $2$ damaged tomatoes, giving your answers to $4$ decimal places.
[3]
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