Suppose we have two bags of coins, both with 1p and 2p pieces.

One coin is taken from bag 1 and put into bag 2. Two coins are then taken from bag 2. I want to find the probability distribution for the value of coins taken from bag 2.

The probability that a 2p coin is taken from bag 1 is 3/5. If a 2p coin is taken from bag 1 and put into bag 2, it has 4 2p coins and 2 1p coins.

We can then take from bag 2:

1p and 1p (=2p) with probability

1p and 2p (or 2p and 1p) (=3p) with probability

2p and 2p (=4p) with probability

We can put these values and probabilities in a table.

Value |
2p |
3p |
4p |

Probability |
2/30 |
16/30 |
12/30 |

Because the probability of taking a 2p coin from bag 1 is 3/5 so all these probabilities are multiplied by 3/5.

Value |
2p |
3p |
4p |

Probability |
6/150 |
48/150 |
36/150 |

The probability that a 1p coin is taken from bag 1 is 2/5. If a 1p coin is taken from bag 1 and put into bag 2, it has 3 2p coins and 3 1p coins.

We can then take from bag 2:

1p and 1p (=2p) with probability

1p and 2p (or 2p and 1p) (=3p) with probability

2p and 2p (=4p) with probability

We can put these values and probabilities in a table.

Value |
2p |
3p |
4p |

Probability |
6/30 |
18/30 |
6/30 |

Because the probability of taking a 2p coin from bag 1 is 3/5 so all these probabilities are multiplied by 3/5.

Value |
2p |
3p |
4p |

Probability |
12/150 |
36/150 |
12/150 |

The vales in the highlighted tables are then added to give the overall probability distribution.

Value |
2p |
3p |
4p |

Probability |
18/150 |
84/150 |
48/150 |