Alright, let's dive into this problem step by step! We'll use some probability theory and basic algebra to solve this problem.
We have a random variable X which represents the total dollar value of three bills selected at random from a bag. The bag contains the following:
- Three 20-dollar bills
- One 50-dollar bill
- One 100-dollar bill
We want to find E[X], which is the expected value of X. The expected value is essentially a weighted average; it takes into account all the possible outcomes of the random variable (in this case, the total dollar values) and the probability of each outcome.
Let's first list all the ways we can choose three bills from the bag:
1. Three 20-dollar bills: Total value = $20 + $20 + $20 = $60
2. Two 20-dollar bills and one 50-dollar bill: Total value = $20 + $20 + $50 = $90
3. Two 20-dollar bills and one 100-dollar bill: Total value = $20 + $20 + $100 = $140
4. One 20-dollar bill, one 50-dollar bill, and one 100-dollar bill: Total value = $20 + $50 + $100 = $170
Now let's calculate the probability of each outcome. There are 5 bills in total, and we're choosing 3, so the number of ways to choose 3 bills from 5 is "5 choose 3", which is 5!/(3!2!) = 10.
1. Probability of choosing three 20-dollar bills is (3/5)*(2/4)*(1/3) = 1/10.
2. Probability of choosing two 20s and one 50 is (3/5)*(2/4)*(1/3)*3 = 3/10. (3 because we can choose the 50-dollar bill in three different ways)
3. Probability of choosing two 20s and one 100 is (3/5)*(2/4)*(1/3)*3 = 3/10. (similar to above)
4. Probability of choosing one 20, one 50, and one 100 is (3/5)*(1/4)*(1/3)*3 = 3/10. (3 because we can choose the 20-dollar bill in three different ways)
Finally, let's compute the expected value:
E[X] = (1/10)*60 + (3/10)*90 + (3/10)*140 + (3/10)*170
= 6 + 27 + 42 + 51
= 126.
So, the expected total dollar value of the selected bills is $126.