Question

A change machine is broken. If you put in a penny, it spits out a coin at random: either a penny, nickel, dime, or quarter, each with equal probability. How much profit can you expect to make by inserting 100 pennies?

Answer 1

By calculating the expected value, we find that inserting 100 pennies into the broken change machine would result in an expected profit of $9.25, assuming an equal chance of getting a penny, nickel, dime, or quarter back for each penny inserted.

Step-by-step explanation:

To calculate the expected profit from inserting 100 pennies into the broken change machine, we first need to understand the concept of expected value. Each time you insert a penny, you have an equal probability of getting back a penny, nickel (5 cents), dime (10 cents), or quarter (25 cents). Since each outcome is equally likely, the expected value for a single penny is calculated by averaging these outcomes.

Expected value for one penny = (1 + 5 + 10 + 25) / 4 = 41 / 4 = 10.25 cents.

This means on average, you gain 10.25 cents - 1 cent (the cost of the penny you inserted) = 9.25 cents per penny inserted.

If you insert 100 pennies, the expected profit would be 100 times the profit per penny:

Expected profit = 100 * 9.25 cents = 925 cents or $9.25