Question
At a local school, some percentage of seniors attend prom. Only 61% of the students go to prom and the senior lock-in. The probability of a student attending senior lock-in given that they attended prom is 0.73. What percentage of seniors attend prom, rounded to the nearest whole percent?
Answers
GPT-4o mini
To find the percentage of seniors who attend prom, we can use the information provided and apply the definitions of conditional probability.
Let:
- \( P(A) \) be the probability of attending prom.
- \( P(B) \) be the probability of attending the senior lock-in.
- \( P(B | A) \) be the probability of attending the senior lock-in given they attended prom.
From the problem, we know:
- \( P(A) = 0.61 \) (61% attend both prom and the senior lock-in)
- \( P(B | A) = 0.73 \)
We can use the law of total probability. We know that if students attend prom, there is a certain probability that they will also attend the lock-in, denoted as \( P(B | A) \). We can express the probability of students attending the senior lock-in as follows:
\[
P(B) = P(B | A) \cdot P(A)
\]
Substituting the values we have:
\[
P(B) = 0.73 \cdot 0.61
\]
\[
P(B) = 0.4453
\]
Since we know \( P(B) \) (the total percentage of students attending the senior lock-in) is not provided, we focus on the percentage of seniors attending prom based solely on the percentage of students who attend both events.
Using the conditional probability interpretation;
Let \( p \) be the percentage of seniors attending prom.
Since \( P(B | A) = 0.73 \), we express the total number of prom-goers who also go to senior lock-in:
\[
\text{Total students attending prom and lock-in} = P(A) \cdot P(B|A) = 0.61p
\]
However, we only need \( p \), and we already know \( 0.73\) of those attending prom attend lock-in, which doesn't input directly into \( p \). So we can't directly infer \( p \) without more details about the overall population of seniors.
For simplification—and if all students attending prom are assumed to be seniors given in our problem—only those that fit these percentages matter out of seniors.
Since it is not clear otherwise, we assume that all seniors make up the entire student body attending prom, hence:
The percentage of seniors attending prom = \( p \).
To confirm:
Assuming from simplistically \( p = \frac{P(A)}{P(B | A)} \)
\[
p = \frac{0.61}{0.73} \approx 0.8356 \approx 83.6\%
\]
Thus rounding off gives that a total of **84%** of seniors attend prom summarized and analyzed correctly thereby.
Thus, the answer is:
\[
\boxed{84}
\]
This establishes the effective metrics needed around the gathered result of prom attendance defined distinctly via calculated competencies.
Let:
- \( P(A) \) be the probability of attending prom.
- \( P(B) \) be the probability of attending the senior lock-in.
- \( P(B | A) \) be the probability of attending the senior lock-in given they attended prom.
From the problem, we know:
- \( P(A) = 0.61 \) (61% attend both prom and the senior lock-in)
- \( P(B | A) = 0.73 \)
We can use the law of total probability. We know that if students attend prom, there is a certain probability that they will also attend the lock-in, denoted as \( P(B | A) \). We can express the probability of students attending the senior lock-in as follows:
\[
P(B) = P(B | A) \cdot P(A)
\]
Substituting the values we have:
\[
P(B) = 0.73 \cdot 0.61
\]
\[
P(B) = 0.4453
\]
Since we know \( P(B) \) (the total percentage of students attending the senior lock-in) is not provided, we focus on the percentage of seniors attending prom based solely on the percentage of students who attend both events.
Using the conditional probability interpretation;
Let \( p \) be the percentage of seniors attending prom.
Since \( P(B | A) = 0.73 \), we express the total number of prom-goers who also go to senior lock-in:
\[
\text{Total students attending prom and lock-in} = P(A) \cdot P(B|A) = 0.61p
\]
However, we only need \( p \), and we already know \( 0.73\) of those attending prom attend lock-in, which doesn't input directly into \( p \). So we can't directly infer \( p \) without more details about the overall population of seniors.
For simplification—and if all students attending prom are assumed to be seniors given in our problem—only those that fit these percentages matter out of seniors.
Since it is not clear otherwise, we assume that all seniors make up the entire student body attending prom, hence:
The percentage of seniors attending prom = \( p \).
To confirm:
Assuming from simplistically \( p = \frac{P(A)}{P(B | A)} \)
\[
p = \frac{0.61}{0.73} \approx 0.8356 \approx 83.6\%
\]
Thus rounding off gives that a total of **84%** of seniors attend prom summarized and analyzed correctly thereby.
Thus, the answer is:
\[
\boxed{84}
\]
This establishes the effective metrics needed around the gathered result of prom attendance defined distinctly via calculated competencies.