Questions: A test was given to a group of students. The grades and gender are summarized below A B C Total ----------------------- Male 15 18 19 52 Female 20 10 12 42 Total 35 28 31 94 If one student is chosen at random from those who took the test, find the probability that the student was female GIVEN they got a 'C'.

A test was given to a group of students. The grades and gender are summarized below A B C Total ----------------------- Male 15 18 19 52 Female 20 10 12 42 Total 35 28 31 94 If one student is chosen at random from those who took the test, find the probability that the student was female GIVEN they got a 'C'.
Transcript text: A test was given to a group of students. The grades and gender are summarized below \begin{tabular}{|r|r|r|r|r|} \hline & A & B & C & Total \\ \hline Male & 15 & 18 & 19 & 52 \\ \hline Female & 20 & 10 & 12 & 42 \\ \hline Total & 35 & 28 & 31 & 94 \\ \hline \end{tabular} If one student is chosen at random from those who took the test, Find the probability that the student was female GIVEN they got a ' C '.
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Solution

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Solution Steps

Step 1: Define the Problem

We need to find the probability that a student was female given that they received a grade of 'C'. This can be expressed mathematically using conditional probability:

\[ P(\text{Female} \mid \text{C}) = \frac{P(\text{Female and C})}{P(\text{C})} \]

Step 2: Gather Data

From the provided table, we extract the following values:

  • Number of females who got a 'C': \( 12 \)
  • Total number of students who got a 'C': \( 31 \)
  • Total number of students: \( 94 \)
Step 3: Calculate Probabilities

We calculate the probabilities needed for the conditional probability formula:

  1. The probability of a student being female and getting a 'C': \[ P(\text{Female and C}) = \frac{12}{94} \]

  2. The probability of a student getting a 'C': \[ P(\text{C}) = \frac{31}{94} \]

Step 4: Compute Conditional Probability

Now we can substitute these probabilities into the conditional probability formula:

\[ P(\text{Female} \mid \text{C}) = \frac{\frac{12}{94}}{\frac{31}{94}} = \frac{12}{31} \]

Calculating this gives:

\[ P(\text{Female} \mid \text{C}) \approx 0.3871 \]

Final Answer

The probability that the student was female given they got a 'C' is:

\[ \boxed{0.3871} \]

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