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Chapter 16: Problem 57

A box contains twenty \(\$ 1\) bills, ten \(\$ 5\) bills, five \(\$ 10\) bills, four \(\$ 20\) bills, and one \(\$ 100\) bill. You blindly reach into the box and draw a bill at random. What is the expected value of your draw?

Short Answer

Expert verified
The expected value of your draw is \$7.5

Step by step solution

01

Determine the total number of bills

Add the number of each type of bill together to find the total number of bills in the box, which is \(20 + 10 + 5 + 4 + 1 = 40\).
02

Calculate the Individual Probabilities

The probability of drawing a type of bill equals to the number of that type of bill divided by the total number of bills. So, the probability of drawing each dollar bill respectively is, for \$1 it's \(20/40 = 0.5\), for \$5 it's \(10/40 = 0.25\), for \$10 it's \(5/40 = 0.125\), for \$20 it's \(4/40 = 0.1\), and for \$100 it's \(1/40 = 0.025\).
03

Calculate the Expected Value

The expected value is calculated by multiplying each possible outcome by their respective probabilities, and then adding these values up. So it's \((1 * 0.5) + (5 * 0.25) + (10 * 0.125) + (20 * 0.1) + (100 * 0.025) = 0.5 + 1.25 + 1.25 + 2.0 + 2.5 = 7.5\)

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