From simple questions to complex issues, IDNLearn.com has the answers you need. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.
Sagot :
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]
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]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Accurate answers are just a click away at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.