Question: Giving a test to a group of students, the grades and genders are summarized below.
Male 2 11 16 29
Female 19 6 20 45
Total 21 17 36 74
If one student is chosen at random, find the probability that the student was female OR received a grade of "C". (Round answer to four decimal places.)
1. **State the problem:** We want to find the probability that a randomly chosen student is either female or received a grade of C.
2. **Identify the data from the contingency table:**
- Total students: $74$
- Number of females: $45$
- Number of students who received grade C: $36$
- Number of females who received grade C: $20$
3. **Recall the formula for the probability of A or B:**
$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
where
- $A$ = event student is female
- $B$ = event student received grade C
4. **Calculate each probability:**
$$P(\text{female}) = \frac{45}{74}$$
$$P(\text{grade C}) = \frac{36}{74}$$
$$P(\text{female and grade C}) = \frac{20}{74}$$
5. **Apply the formula:**
$$P(\text{female or grade C}) = \frac{45}{74} + \frac{36}{74} - \frac{20}{74} = \frac{45 + 36 - 20}{74} = \frac{61}{74}$$
6. **Simplify the fraction:**
$$\frac{61}{74}$$ cannot be simplified further.
7. **Convert to decimal and round to four decimal places:**
$$\frac{61}{74} \approx 0.8243$$
**Final answer:**
The probability that the student was female or received a grade of C is approximately $0.8243$.