Let's solve this problem step-by-step to determine the probability of drawing a chocolate chip cookie first (Event A) and then an oatmeal raisin cookie second (Event B).
1. Calculate the Probability of Event A:
Event [tex]$A$[/tex] is the event of drawing a chocolate chip cookie from the jar on the first draw. The cookie jar contains:
- 3 chocolate chip cookies
- 4 oatmeal raisin cookies
- 5 sugar cookies
Therefore, the total number of cookies in the jar is:
[tex]\[
3 + 4 + 5 = 12
\][/tex]
The probability of drawing a chocolate chip cookie on the first draw is:
[tex]\[
P(A) = \frac{\text{Number of chocolate chip cookies}}{\text{Total number of cookies}} = \frac{3}{12} = \frac{1}{4}
\][/tex]
2. Calculate the Probability of Event B given Event A:
Event [tex]$B|A$[/tex] is the event of drawing an oatmeal raisin cookie on the second draw, given that the first cookie drawn was a chocolate chip cookie.
After drawing one chocolate chip cookie, there are:
- 2 chocolate chip cookies left
- 4 oatmeal raisin cookies
- 5 sugar cookies
This means there are now a total of:
[tex]\[
12 - 1 = 11 \text{ cookies remaining}
\][/tex]
The probability of drawing an oatmeal raisin cookie after one chocolate chip cookie has been drawn is:
[tex]\[
P(B|A) = \frac{\text{Number of oatmeal raisin cookies}}{\text{Remaining total number of cookies}} = \frac{4}{11}
\][/tex]
3. Calculate the Joint Probability [tex]$P(A \text{ and } B$[/tex]):
The probability of both events happening (drawing a chocolate chip cookie first and then an oatmeal raisin cookie) is calculated by multiplying the probability of Event [tex]$A$[/tex] by the conditional probability of Event [tex]$B$[/tex] given Event [tex]$A$[/tex]:
[tex]\[
P(A \text{ and } B) = P(A) \times P(B|A)
\][/tex]
[tex]\[
P(A \text{ and } B) = \frac{1}{4} \times \frac{4}{11} = \frac{4}{44} = \frac{1}{11}
\][/tex]
Therefore, the probability of drawing a chocolate chip cookie on the first draw followed by an oatmeal raisin cookie on the second draw is [tex]\(\frac{1}{11}\)[/tex].
The correct answer is:
B) [tex]\(\frac{1}{11}\)[/tex]
