Math Problem Statement

Find the probability that a student was female, the probability of being female and getting an A, and other related conditional probabilities based on a table of student grades and gender.

Solution

Let's work through the probability questions from the table in your image.

Given table:

ABCTotal
Male420832
Female135725
Total17251557

We will solve each question one by one:

A. Find the probability that the student was female.

The total number of students is 57. The number of females is 25.

P(\text{female}) = \frac{\text{Number of females}}{\text{Total students}} = \frac{25}{57} \approx 0.4386 \quad \text{(or 43.86%)}.

Your input of 26.50% seems to have been incorrect.

B. Find the probability that the student was female and got an "A."

The number of females who got an "A" is 13. The total number of students is 57.

P(\text{female AND A}) = \frac{\text{Number of females who got an A}}{\text{Total students}} = \frac{13}{57} \approx 0.2281 \quad \text{(or 22.81%)}.

Your input of 18.39% was incorrect as well.

C. Find the probability that the student was female or got an "A."

To calculate the probability of female OR A, we apply the inclusion-exclusion principle:

P(female OR A)=P(female)+P(A)P(female AND A).P(\text{female OR A}) = P(\text{female}) + P(\text{A}) - P(\text{female AND A}).

  • P(female)=25570.4386P(\text{female}) = \frac{25}{57} \approx 0.4386,
  • P(A)=17570.2982P(\text{A}) = \frac{17}{57} \approx 0.2982,
  • P(female AND A)=13570.2281P(\text{female AND A}) = \frac{13}{57} \approx 0.2281.

Now, substitute these into the formula:

P(\text{female OR A}) = 0.4386 + 0.2982 - 0.2281 = 0.5087 \quad \text{(or 50.87%)}.

D. If one student is chosen at random, find the probability that the student was female given they got an "A."

This is a conditional probability question:

P(female | A)=P(female AND A)P(A).P(\text{female | A}) = \frac{P(\text{female AND A})}{P(\text{A})}.

From earlier, we know:

  • P(female AND A)=13570.2281P(\text{female AND A}) = \frac{13}{57} \approx 0.2281,
  • P(A)=17570.2982P(\text{A}) = \frac{17}{57} \approx 0.2982.

Now calculate the conditional probability:

P(\text{female | A}) = \frac{0.2281}{0.2982} \approx 0.7651 \quad \text{(or 76.51%)}.

Your input of 76.47% was very close!


Would you like more details on any of the calculations? Here are 5 follow-up questions to deepen your understanding:

  1. How does the inclusion-exclusion principle work in probability?
  2. What is the difference between conditional and joint probability?
  3. Can you apply these concepts to a different set of data?
  4. How would the results change if more students were added to the dataset?
  5. Why is it important to use total numbers in probability questions like these?

Tip: Always double-check probabilities to ensure they add up logically—this can prevent calculation errors!

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Math Problem Analysis

Mathematical Concepts

Probability
Conditional Probability
Inclusion-Exclusion Principle

Formulas

P(female) = Number of females / Total students
P(female AND A) = Number of females who got an A / Total students
P(female OR A) = P(female) + P(A) - P(female AND A)
P(female | A) = P(female AND A) / P(A)

Theorems

Basic Probability
Inclusion-Exclusion Principle
Conditional Probability Theorem

Suitable Grade Level

High School - Grades 9-12