(a) Suppose you have two quarters and a dime in your left pocket and two dimes and three quarters in your right pocket. You select a pocket at random and from it a coin at random. What is the probability that it is a dime? (b) Let \(x\) be the amount of money you select. Find \(E(x)\). (c) Suppose you selected a dime in (a). What is the probability that it came from your right pocket? (d) Suppose you do not replace the dime, but select another coin which is also a dime. What is the probability that this second coin came from your right pocket?

Random selection is a fundamental aspect of probability theory. It means selecting items in such a way that every item has an equal chance of being chosen.

In this exercise, you start by selecting a pocket at random. Since you have two pockets and you don't favor one over the other, the probability of choosing either pocket is the same. This can be expressed mathematically as:

\( P(Choosing\text{ }Left\text{ }Pocket) = P(Choosing\text{ }Right\text{ }Pocket) = \frac{1}{2} \).

After choosing a pocket, you then select a coin at random. The concept of random selection ensures that each coin in the chosen pocket has an equal chance of being picked. For example, in the left pocket, where there are 3 coins (2 quarters and 1 dime), the probability of picking any one coin is \( \frac{1}{3} \) .

• From the left pocket, the chance of selecting the dime is \( \frac{1}{3} \).
• From the right pocket, where there are 5 coins (2 dimes and 3 quarters), the probability of selecting a dime is \( \frac{2}{5} \).

Expected value is a concept in probability that provides a measure of the center of a probability distribution. In simpler terms, it is the average outcome if an experiment is repeated many times.

To find the expected value in this exercise, consider the amount of money you might select from either pocket:

• In the left pocket, you have two quarters and one dime. The expected value from the left pocket can be calculated as follows:
  
  \[ E(x\text{ if selected from left pocket}) = \frac{2 \times 25 + 1 \times 10}{2 + 1} = \frac{60}{3} = 20\text{ cents} = 0.20\text{ dollars}. \]
• In the right pocket, you have three quarters and two dimes. The expected value from the right pocket can be calculated as:
  
  \[ E(x\text{ if selected from right pocket}) = \frac{3 \times 25 + 2 \times 10}{3 + 2} = \frac{55}{5} = 11\text{ cents} = 0.11\text{ dollars}. \]

The overall expected value, considering that each pocket is equally likely to be chosen, is calculated as: \[ E(x) = \frac{1}{2} \times 0.20 + \frac{1}{2} \times 0.11 = 0.155\text{ dollars}. \]

This means that, on average, you can expect to select about 15.5 cents.

Bayes' Theorem provides a way to update probabilities based on new information. It is extremely useful in determining conditional probabilities.

In this exercise, after determining the probability of selecting a dime, you use Bayes' Theorem to find the probability that the dime came from the right pocket. The formula for Bayes' Theorem is: \[ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}. \]

Here, B is the event of selecting a dime, and A is the event that the dime came from the right pocket:

\[ P(Right | Dime) = \frac{P(Dime | Right) \times P(Right)}{P(Dime)}. \]

Substituting the known probabilities:

\[ P(Right | Dime) = \frac{(\frac{2}{5}) \times (\frac{1}{2})}{\frac{11}{30}} = \frac{15}{22}. \]

This means that if you have already selected a dime, there is a 15/22 chance that it came from the right pocket.

Furthermore, suppose you do not replace the dime and you select another coin which is also a dime. To find the probability that this second dime came from your right pocket, you again use Bayes' Theorem:

• Recall that probability of selecting the second dime from the right pocket is \ (\frac{1}{4}) \.
• The updated probability after selecting the first dime is:

\[ P(Right | Second Dime) = \frac{P(Second Dime | Right) \times P(Right | First Dime)}{P(Second Dime)} = \frac{(15/22) \times (\frac{1}{4})}{\text{Calculation from Step 4}} = \frac{5}{6}. \]

This updated probability takes into account the reduced number of coins and the conditional nature of the selection process.