## Probability Distribution For the Value of Coins When Coins Are Taken From One Bag and Put Into Another

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

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

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

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 1is 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 is2/5. If a 1p coin is taken from bag 1 and put into bag 2, it has 3 2pcoins 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 1is 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 togive the overall probability distribution.

Value | 2p | 3p | 4p |

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