Sure, let's solve this step-by-step.
1. Total number of cookies:
Gwen's bag contains the following cookies:
- Chocolate chip: 5
- Peanut butter: 8
- Sugar: 7
- Oatmeal raisin: 8
Therefore, the total number of cookies is:
[tex]\[
5 + 8 + 7 + 8 = 28
\][/tex]
2. Probability of selecting the first peanut butter cookie:
Since there are 8 peanut butter cookies out of 28 total cookies, the probability of selecting one peanut butter cookie is:
[tex]\[
\frac{8}{28} = \frac{2}{7}
\][/tex]
3. Adjusting the number of cookies after one is eaten:
If Gwen eats the peanut butter cookie she selected, the number of peanut butter cookies reduces to 7, and the total number of cookies reduces to 27.
4. Probability of selecting the second peanut butter cookie:
Now, there are 7 peanut butter cookies left out of the 27 cookies. The probability of selecting another peanut butter cookie is:
[tex]\[
\frac{7}{27}
\][/tex]
5. Combined probability of both events:
To find the probability that both events occur (selecting and eating the first peanut butter cookie, and then selecting another peanut butter cookie), we multiply the probabilities of each event:
[tex]\[
\left(\frac{2}{7}\right) \times \left(\frac{7}{27}\right) = \frac{2 \times 7}{7 \times 27} = \frac{2}{27}
\][/tex]
Thus, the probability that Gwen randomly selects a peanut butter cookie, eats it, then randomly selects another peanut butter cookie is:
[tex]\[
\boxed{\frac{2}{27}}
\][/tex]
