Multiple independent coins/dice

5 Harry has three coins.

• One coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 3 }\).
• The second coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 4 }\).
• The third coin is biased so that, when it is thrown, the probability of obtaining a head is \(\frac { 1 } { 5 }\).

The random variable \(X\) is the number of heads that Harry obtains when he throws all three coins together.

1. Find the probability generating function of \(X\).
   Isaac has two fair coins. The random variable \(Y\) is the number of heads that Isaac obtains when he throws both of his coins together. The random variable \(Z\) is the total number of heads obtained when Harry throws his three coins and Isaac throws his two coins.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial in \(t\).
3. Use the probability generating function of \(Z\) to find \(E ( Z )\).

5 Keira has two unbiased coins. She tosses both coins. The number of heads obtained by Keira is denoted by \(X\).

1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
   Hassan has three coins, two of which are biased so that the probability of obtaining a head when the coin is tossed is \(\frac { 1 } { 3 }\). The corresponding probability for the third coin is \(\frac { 1 } { 4 }\). The number of heads obtained by Hassan when he tosses these three coins is denoted by \(Y\).
2. Find the probability generating function \(\mathrm { G } _ { Y } ( \mathrm { t } )\) of \(Y\).
   The random variable \(Z\) is the total number of heads obtained by Keira and Hassan.
3. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
4. Use the probability generating function of \(Z\) to find \(\mathrm { E } ( Z )\).
5. Use the probability generating function of \(Z\) to find the most probable value of \(Z\).

4 Jason has three biased coins. For each coin the probability of obtaining a head when it is thrown is \(\frac { 2 } { 3 }\). Jason throws all three coins. The number of heads obtained is denoted by \(X\).

1. Find the probability generating function \(\mathrm { G } _ { \mathrm { X } } ( \mathrm { t } )\) of \(X\).
   Jason also has two unbiased coins. He throws all five coins. The number of heads obtained from the two unbiased coins is denoted by \(Y\). It is given that \(G _ { Y } ( t ) = \frac { 1 } { 4 } + \frac { 1 } { 2 } t + \frac { 1 } { 4 } t ^ { 2 }\). The random variable \(Z\) is the total number of heads obtained when Jason throws all five coins.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
3. Find \(\mathrm { E } ( \mathrm { Z } )\).

5 A 6 -sided dice, \(A\), with faces numbered \(1,2,3,4,5,6\) is biased so that the probability of throwing a 6 is \(\frac { 1 } { 4 }\). The random variable \(X\) is the number of 6s obtained when dice \(A\) is thrown twice.

1. Find the probability generating function of \(X\).
   A second dice, \(B\), with faces numbered \(1,2,3,4,5,6\) is unbiased. The random variable \(Y\) is the number of 6s obtained when dice \(B\) is thrown twice. The random variable \(Z\) is the total number of 6s obtained when both dice are thrown twice.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
3. Find \(\operatorname { Var } ( Z )\).
4. Use the probability generating function of \(Z\) to find the most probable value of \(Z\).

5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 3 }\) and the third coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 5 }\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.

1. Find the probability generating function of \(X\).
   Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice. The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
   

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3. Use your answer to part (b) to find \(\mathrm { E } ( Z )\).

5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 3 }\) and the third coin is biased so that the probability of obtaining a head is \(\frac { 1 } { 5 }\). The random variable \(X\) is the number of heads that Nikita obtains when he throws all three coins at the same time.

1. Find the probability generating function of \(X\).
   Rajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable \(Y\) is the number of 4 s that Rajesh obtains when he throws the two dice. The random variable \(Z\) is the sum of the number of heads obtained by Nikita and the number of 4 s obtained by Rajesh.
2. Find the probability generating function of \(Z\), expressing your answer as a polynomial.
   

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3. Use your answer to part (b) to find \(\mathrm { E } ( Z )\).

6 Andrew has five coins. Three of them are unbiased. The other two are biased such that the probability of obtaining a head when one of them is tossed is \(\frac { 3 } { 5 }\). Andrew tosses all five coins. It is given that the probability generating function of \(X\), the number of heads obtained on the unbiased coins, is \(\mathrm { G } _ { X } ( t )\), where $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 8 } + \frac { 3 } { 8 } t + \frac { 3 } { 8 } t ^ { 2 } + \frac { 1 } { 8 } t ^ { 3 }$$

1. Find \(G _ { Y } ( \mathrm { t } )\), the probability generating function of \(Y\), the number of heads on the biased coins.
2. The random variable \(Z\) is the total number of heads obtained when Andrew tosses all five coins. Find the probability generating function of \(Z\), giving your answer as a polynomial.
3. Find \(\mathrm { E } ( Z )\) and \(\operatorname { Var } ( Z )\).
4. Write down the value of \(\mathrm { P } ( Z = 3 )\).