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.

Question

What is the expected value of the number shown if we add two additional $2$'s (instead of just one) to the bag?

Answer 1

To find the expected value when we add two slips with the number $2$ to the existing slips in the bag, we first need to determine the total number of slips after the addition and then compute the expected value based on the new composition.

Initially, the bag contains:

• 8 slips with the number $7$
• 2 slips with the number $9$

This gives us a total of:

$$8+2=10\text{ slips}$$

When we add 2 slips with the number $2$, the new composition of slips becomes:

• 8 slips with the number $7$
• 2 slips with the number $9$
• 2 slips with the number $2$

Thus, the total number of slips is now:

$$8+2+2=12\text{ slips}$$

Next, we can calculate the contributions of each number to the expected value. The expected value $E$ is calculated using the formula for the expected value of a discrete random variable, which is:

$$E=\sum (x_i\cdot P(x_i))$$

where $x_i$ are the values and $P(x_i)$ are their respective probabilities.

Now, we find the contributions of each number to the expected value:

1. For the slips with $7$:

   • Number of slips: $8$
   • Contribution: $7\cdot \frac{8}{12}=\frac{56}{12}$

2. For the slips with $9$:

   • Number of slips: $2$
   • Contribution: $9\cdot \frac{2}{12}=\frac{18}{12}$

3. For the slips with $2$:

   • Number of slips: $2$
   • Contribution: $2\cdot \frac{2}{12}=\frac{4}{12}$

Now we sum these contributions:

$$E=\frac{56}{12}+\frac{18}{12}+\frac{4}{12}=\frac{56+18+4}{12}=\frac{78}{12}=6.5$$

Thus, the expected value of the number shown when we add two additional $2$s to the bag is:

$$\boxed{{\displaystyle 6.5}}$$