Question

Suppose we have a bag with $10$ slips of paper in it. Eight slips have a $7$ on them and the other two have a $9$ on them.
What is the expected value of the number shown when we draw a single slip of paper?

Answer 1

To find the expected value of the number shown when drawing a slip of paper from the bag, we can use the formula for expectation, which is defined as:

\[
E[X] = \sum (x_i \cdot P(x_i))
\]

where \( x_i \) is a possible outcome and \( P(x_i) \) is the probability of that outcome.

In our case, we have two different outcomes from the slips of paper: the numbers \( 7 \) and \( 9 \).

1. **Calculate the probabilities of each outcome:**
- There are \( 8 \) slips with a \( 7 \).
- There are \( 2 \) slips with a \( 9 \).
- The total number of slips is \( 10 \).

The probabilities for each outcome are as follows:
- \( P(7) = \frac{8}{10} = 0.8 \)
- \( P(9) = \frac{2}{10} = 0.2 \)

2. **Calculate the expected value:**
Now we can compute the expected value using the probabilities and the outcomes:

\[
E[X] = (7 \cdot P(7)) + (9 \cdot P(9))
\]

Substituting the values:

\[
E[X] = (7 \cdot 0.8) + (9 \cdot 0.2)
\]

Calculating each term:

- \( 7 \cdot 0.8 = 5.6 \)
- \( 9 \cdot 0.2 = 1.8 \)

Now we add these two results:

\[
E[X] = 5.6 + 1.8 = 7.4
\]

Therefore, the expected value of the number shown when we draw a single slip of paper is

\[
\boxed{7.4}
\]