Respuesta :
Answer:
The probability that there is a silver coin in the other drawer is [tex]\frac{2}{3}[/tex]
Step-by-step 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, [tex]P(A)=P(B)=\frac{1}{2}[/tex]
Cabinet A contains a silver coin in each drawer,
i.e. [tex]P(E|A)=1[/tex]
Cabinet B contains a silver coin in one of its drawers and a gold coin in the other,
i.e. [tex]P(E|B)=\frac{1}{2}[/tex]
The probability that there is a silver coin in the other drawer is given by,
Applying Bayes theorem,
[tex]P(A|E)=\frac{P(E|A)P(A)}{P(E|A)P(A)+P(E|B)P(B)}[/tex]
[tex]P(A|E)=\frac{(1)(\frac{1}{2})}{(1)(\frac{1}{2})+(\frac{1}{2})(\frac{1}{2})}[/tex]
[tex]P(A|E)=\frac{\frac{1}{2}}{\frac{1}{2}+\frac{1}{4}}[/tex]
[tex]P(A|E)=\frac{\frac{1}{2}}{\frac{3}{4}}[/tex]
[tex]P(A|E)=\frac{2}{3}[/tex]
Therefore, the probability that there is a silver coin in the other drawer is [tex]\frac{2}{3}[/tex]
