2015 TJC P2 Q9 [Modified]

When a large number of mangoes are harvested, \(37\)% of them are unripe. The mangoes are packed randomly in boxes of \(6\).

1. Show that the probability that there is no unripe mango in a box is \(0.062524\), correct to \(5\) significant figures.[2]
2. Find the probability that there are exactly \(2\) unripe mangoes in \(2\) boxes.[2]

A supermarket orders \(50\) boxes of mangoes daily.

3. Find the probability that there are at least \(9\) boxes with no unripe mango in a particular day.[3]
4. When the supermarket places a daily order for \(10\) weeks, estimate the probability that the mean number of boxes with no unripe mango in a day is between \(3\) and \(9\).[4]