Question

Each of 2 cabinets identical in appearance has 2 drawers. Cabinet A contains a silver coin in each drawer, and cabinet B contains a silver coin in one of its drawers and a gold coin in the other. A cabinet is randomly selected, one of its drawers is opened, and a silver coin is found. What is the probability that there is a silver coin in the other drawer?

Answer 1

The probability that there is a silver coin in the other drawer is
`(2)/(3)`

Explanation:

Given : Each of 2 cabinets identical in appearance has 2 drawers. Cabinet A contains a silver coin in each drawer, and cabinet B contains a silver coin in one of its drawers and a gold coin in the other. A cabinet is randomly selected, one of its drawers is opened, and a silver coin is found.

To find : What is the probability that there is a silver coin in the other drawer?

Solution :

Let A be the event that cabinet A is chosen.

Let B be the event that cabinet B is chosen

Let E be the event that silver coin is chosen.

Each of 2 cabinets identical in appearance has 2 drawers.

So,
`P(A)=P(B)=(1)/(2)`

Cabinet A contains a silver coin in each drawer,

i.e.
`P(E|A)=1`

Cabinet B contains a silver coin in one of its drawers and a gold coin in the other,

i.e.
`P(E|B)=(1)/(2)`

The probability that there is a silver coin in the other drawer is given by,

Applying Bayes theorem,

`P(A|E)=(P(E|A)P(A))/(P(E|A)P(A)+P(E|B)P(B))`

`P(A|E)=((1)((1)/(2)))/((1)((1)/(2))+((1)/(2))((1)/(2)))`

`P(A|E)=((1)/(2))/((1)/(2)+(1)/(4))`

`P(A|E)=((1)/(2))/((3)/(4))`

`P(A|E)=(2)/(3)`

Therefore, the probability that there is a silver coin in the other drawer is
`(2)/(3)`