2015 IJC P2 Q7 [Modified]

Bernard is carrying out an experiment with a fair tetrahedral die, which has its four triangular faces numbered from '\(6\)' to '\(9\)', and a biased \(10\)-sided die numbered from '\(1\)' to '\(10\)'.

1. Bernard rolls the fair die \(9\) times. Find the probability that the die shows a '\(8\)' between \(3\) and \(7\) times, inclusive.
2. Bernard now rolls the fair die \(65\) times. Find the probability that the die shows a '\(9\)' more than \(12\) times.

The probability that the biased die shows a '\(9\)' is \(\frac{1}{25}\).

3. Bernard rolls the biased die \(65\) times.
   1. Find the probability that the biased die shows a '\(9\)' more than \(5\) times.
   2. Bernard repeated the experiment 30 times. Estimate the probability that the average number of \(9\)' shown is at least \(2.1\).