Question 32·Medium·Probability and Conditional Probability
At a certain high school with 120 seniors, 70 are enrolled in AP Calculus, 50 are enrolled in AP Statistics, and 30 are enrolled in both courses.
If a senior is chosen at random from those enrolled in AP Calculus, what is the probability that the student is also enrolled in AP Statistics?
For conditional probability questions, pay close attention to the phrase after words like "given that" or "from those"—this tells you the denominator, the group you are choosing from. Then find how many in that group also satisfy the second condition for the numerator. Write probability as (desired subgroup)/(given group) and simplify the fraction to match the answer choices, ignoring any extra numbers (like the total number of students) that are not part of the given group.
Hints
Focus on the group you are choosing from
The student is chosen "from those enrolled in AP Calculus". Which number from the problem tells you how many students are in that group?
Find how many fit both conditions
Among the AP Calculus students, we want those who are also in AP Statistics. Which number in the problem represents students taking both classes?
Form and simplify the fraction
Put the number in both classes over the number in AP Calculus to form a fraction, then reduce it to lowest terms to match one of the answer choices.
Desmos Guide
Compute the conditional probability as a fraction
In Desmos, type 30/70 or 30 ÷ 70 on a new line. This gives the decimal value of the probability that a randomly chosen AP Calculus student is also in AP Statistics.
Match the fraction to an answer choice
To compare with the options, type 3/7, 1/2, 2/5, and 5/12 on separate lines. Check which of these has the same decimal value as 30/70; that choice is the correct probability.
Step-by-step Explanation
Identify the relevant group (denominator)
The question says: "If a senior is chosen at random from those enrolled in AP Calculus..."
That means our sample space (denominator of the probability) is only the students in AP Calculus.
- Number of seniors in AP Calculus: .
So the denominator of our probability fraction will be .
Identify the desired subgroup (numerator)
We want the probability that a randomly chosen Calculus student is also in AP Statistics.
That means we are interested in students who are in both AP Calculus and AP Statistics.
- Number of seniors in both courses: .
So the numerator of our probability fraction will be .
Write and simplify the conditional probability
The probability that a randomly chosen AP Calculus student is also in AP Statistics is
Now simplify by dividing numerator and denominator by :
So the correct answer is , which corresponds to choice B.