Question

Suppose you have 3 nickels and 4 dimes in your right pocket and 2 nickels and a quarter in your left pocket. You pick a pocket at random and from it select a coin at random. If it is a nickel, what is the probability that it came from your right pocket?

Short answer

Answer

The probability of the nickel coming from the right pocket is $\frac{9}{23}$.

Step-by-step solution

Given Information

There are 3 nickels and 4 dimes in the right pocket and 2 nickels and a quarter in the left pocket

Definition of Independent Event

The events are said to be independent when the occurrence or non-occurrence of any event does not have any effect on the occurrence or non-occurrence of the other event.

Finding the probability that of picking a nickel

Let A be the event that a nickel is picked.

There are total 7 coins in right pocket out of which 3 are nickels and total 3 coins in the left pocket out of which 2 are nickels.

Find the probability of picking a nickel from any pocket.

$$\begin{gathered} P\left(A\right)=\frac{3}{4+3}+\frac{2}{2+1} \\ =\frac{3}{7}+\frac{2}{3} \\ =\frac{23}{21} \end{gathered}$$

Finding the probability of picking the nickel from right pocket.

Find the probability of picking the nickel from right pocket using the Bayes theorem, $P_A\left(B\right)=\frac{A\left(AB\right)}{P\left(A\right)}$.

role="math" localid="1654837853424" $$\begin{gathered} P_A\left(B\right)=\frac{{\displaystyle \frac{3}{7}}}{{\displaystyle \frac{23}{21}}} \\ =\frac{9}{23} \end{gathered}$$

Thus, the desired probability is $\frac{9}{23}$.