Let X represent the outcome of a single roll of a fair die. Suppose you will win $2 if the outcome is either 2 or 5, and lose $1 otherwise. What are your expected winnings in 20 tosses?
To find the expected winnings, we need to calculate the probability and the corresponding winnings for each possible outcome and then multiply them together.
Let's break down the problem into steps:
Step 1: Calculate the probability of rolling a 2 or a 5.
Since we have a fair die with 6 sides, each side has an equal chance of appearing, which means the probability of rolling a specific number is 1/6.
So, the probability of rolling a 2 or a 5 is 1/6 + 1/6 = 1/3.
Step 2: Calculate the winnings for rolling a 2 or a 5.
If we roll a 2 or a 5, we win $2. So the winnings for each favorable outcome are $2.
Step 3: Calculate the probability of not rolling a 2 or a 5.
Since the probability of rolling a 2 or a 5 is 1/3, the probability of not rolling a 2 or a 5 becomes 1 - 1/3 = 2/3.
Step 4: Calculate the winnings for not rolling a 2 or a 5.
If we do not roll a 2 or a 5, we lose $1. So the winnings for each unfavorable outcome are -$1.
Step 5: Calculate the expected winnings for a single toss.
The expected winnings for a single toss can be calculated by multiplying each outcome's probability by its corresponding winnings, and then summing them up:
Expected winnings for a single toss = (Probability of winning * Winnings for winning) + (Probability of losing * Winnings for losing)
Expected winnings for a single toss = (1/3 * $2) + (2/3 * -$1) = $2/3 - $2/3 = $0.
Step 6: Calculate the expected winnings for 20 tosses.
The expected winnings for 20 tosses can be calculated by multiplying the expected winnings for a single toss by the number of tosses:
Expected winnings for 20 tosses = Expected winnings for a single toss * 20 = $0 * 20 = $0.
Therefore, the expected winnings in 20 tosses would be $0.