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Question
Math
Posted 10 months ago

Ana is a teacher who plays a review game with her class. The game involves writing each student's name on an identical slip of paper and selecting students at random. Here's the makeup of her class:
\begin{tabular}{lrrr} 
Grade & 9th 9^{\text {th }} & 10th 10^{\text {th }} & 11th 11^{\text {th }} \\
\hline Number of students & 12 & 9 & 7
\end{tabular}

Suppose that Ana picks a name, replaces it, and picks a name again.
What is the probability that NEITHER of the students selected are 9th 9^{\text {th }} graders?
Round your answer to two decimal places.
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Answer from Sia
Posted 10 months ago
Solution by Steps
step 1
Calculate the total number of students in Ana's class by adding the number of students in each grade
$ \text{Total students} = 12 + 9 + 7 $
step 2
Compute the probability of not picking a 9th grader in one draw
$ P(\text{not 9th grader}) = \frac{\text{Total students} - \text{Number of 9th graders}}{\text{Total students}} $
step 3
Use the total number of students from step 1 to find the probability of not picking a 9th grader
$ P(\text{not 9th grader}) = \frac{28 - 12}{28} $
step 4
Simplify the probability calculated in step 3
$ P(\text{not 9th grader}) = \frac{16}{28} $
step 5
Since Ana replaces the name after each draw, the probability of not picking a 9th grader in the second draw is the same as the first
$ P(\text{not 9th grader second draw}) = P(\text{not 9th grader}) $
step 6
Calculate the probability that neither of the two students picked are 9th graders by multiplying the probabilities of each individual draw
$ P(\text{neither 9th graders}) = P(\text{not 9th grader}) \times P(\text{not 9th grader second draw}) $
step 7
Compute the final probability using the simplified probability from step 4
$ P(\text{neither 9th graders}) = \left(\frac{16}{28}\right) \times \left(\frac{16}{28}\right) $
step 8
Simplify the expression from step 7 to get the final probability
$ P(\text{neither 9th graders}) = \left(\frac{16}{28}\right)^2 $
step 9
Calculate the numerical value of the probability
$ P(\text{neither 9th graders}) = \left(\frac{4}{7}\right)^2 $
step 10
Simplify the numerical value to get the final answer
$ P(\text{neither 9th graders}) = \left(\frac{4}{7}\right)^2 = \frac{16}{49} $
step 11
Round the final answer to two decimal places
$ P(\text{neither 9th graders}) \approx 0.33 $
[1] Answer
P(neither 9th graders)≈0.33 P(\text{neither 9th graders}) \approx 0.33
Key Concept
Independent Events and Replacement
Explanation
The probability of not selecting a 9th grader twice in a row with replacement is the product of the probabilities of not selecting a 9th grader in each individual draw. Since the draws are independent and the name is replaced after each draw, the probability remains the same for each draw.

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