Two-stage binomial problems

4 A fair tetrahedral die has faces numbered \(1,2,3,4\). A coin is biased so that the probability of showing a head when thrown is \(\frac { 1 } { 3 }\). The die is thrown once and the number \(n\) that it lands on is noted. The biased coin is then thrown \(n\) times. So, for example, if the die lands on 3 , the coin is thrown 3 times.

1. Find the probability that the die lands on 4 and the number of times the coin shows heads is 2 .
2. Find the probability that the die lands on 3 and the number of times the coin shows heads is 3 .
3. Find the probability that the number the die lands on is the same as the number of times the coin shows heads.

7. A biased spinner can only land on one of the numbers \(1,2,3\) or 4 . The random variable \(X\) represents the number that the spinner lands on after a single spin and \(\mathrm { P } ( X = r ) = \mathrm { P } ( X = r + 2 )\) for \(r = 1,2\) Given that \(\mathrm { P } ( X = 2 ) = 0.35\)

1. find the complete probability distribution of \(X\). Ambroh spins the spinner 60 times.
2. Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures.