A test was given to a group of students. The grades and residency are summarized below A B C Total In-State 4 9 12 25 Out-of-State 13 20 2 35 Total 17 29 14 60 If one student is chosen at random from those who took the test, find the probability that the student got a C given they are Out-of-State. Write your answer as a reduced fraction.
The Correct Answer and Explanation is:
We are given the following data:
- In-State Students:
- Grade A: 4
- Grade B: 9
- Grade C: 12
- Total In-State: 25
- Out-of-State Students:
- Grade A: 13
- Grade B: 20
- Grade C: 2
- Total Out-of-State: 35
- Total:
- Grade A: 17
- Grade B: 29
- Grade C: 14
- Total Students: 60
We are asked to find the probability that a student got a C given that they are Out-of-State.
Step 1: Identify the conditional probability formula
The probability of getting a C given the student is Out-of-State is a conditional probability. This is given by the formula:P(C∣Out-of-State)=Number of Out-of-State students who got a CTotal number of Out-of-State studentsP(\text{C} \mid \text{Out-of-State}) = \frac{\text{Number of Out-of-State students who got a C}}{\text{Total number of Out-of-State students}}P(C∣Out-of-State)=Total number of Out-of-State studentsNumber of Out-of-State students who got a C
Step 2: Plug in the values
From the data:
- The number of Out-of-State students who got a C = 2
- The total number of Out-of-State students = 35
Thus, the probability is:P(C∣Out-of-State)=235P(\text{C} \mid \text{Out-of-State}) = \frac{2}{35}P(C∣Out-of-State)=352
Step 3: Simplify the fraction
The fraction 235\frac{2}{35}352 is already in its simplest form.
Final Answer:
The probability that a randomly chosen student who is Out-of-State got a C is:235\boxed{\frac{2}{35}}352
Explanation:
To find this probability, we focus on the Out-of-State group and then calculate how many of those students got a C. The formula for conditional probability helps us account for the total number of students in the Out-of-State group, which is 35. Since only 2 students in that group got a C, the probability is simply 235\frac{2}{35}352.
