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

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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