Nested binomial expected count

5 Screws are sold in packets of 15. Faulty screws occur randomly. A large number of packets are tested for faulty screws and the mean number of faulty screws per packet is found to be 1.2 .

1. Show that the variance of the number of faulty screws in a packet is 1.104 .
2. Find the probability that a packet contains at most 2 faulty screws. Damien buys 8 packets of screws at random.
3. Find the probability that there are exactly 7 packets in which there is at least 1 faulty screw.

3 The weights of bags of a particular brand of flour are quoted as 1.5 kg . In fact, on average \(10 \%\) of bags are underweight.

1. Find the probability that, in a random sample of 50 bags, there are exactly 5 bags which are underweight.
2. Bags are randomly chosen and packed into boxes of 20 . Find the probability that there is at least one underweight bag in a box.
3. A crate contains 48 boxes. Find the expected number of boxes in the crate which contain at least one underweight bag.

1. A bag contains 10 counters each with exactly one number written on it.

There are 6 counters with the number 7 on them
There are 3 counters with the number 8 on them
There is 1 counter with the number 9 on it
A random sample of 3 counters is taken from the bag (without replacement).
These counters are then put back in the bag.
This process is then repeated until 20 samples have been taken.
The random variable \(Y\) represents the number of these 20 samples that contain the counter with the number 9 on it.

1. 1. Find the mean of \(Y\)
   2. Find the variance of \(Y\) A random sample of 3 counters is chosen from the bag (without replacement).
2. List all possible samples where the median of the numbers on the 3 counters is 7
3. Find the sampling distribution of the median of the numbers on the 3 counters.