At Mid-State University, 10% of the student body are scholarship athletes. Half of these athletes are left-handers and half are right-handers. Of the students that are not scholarship athletes, 57% are left-handers and 43% are right-handers.

What is the proper probability notation for the statement “half of the scholarship athletes are left-handers”.

Question

At Mid-State University, 10% of the student body are scholarship athletes. Half of these athletes are left-handers and half are right-handers. Of the students that are not scholarship athletes, 57% are left-handers and 43% are right-handers.

What is the proper probability notation for the statement “half of the scholarship athletes are left-handers”.

BlackTom AI · Accepted Answer

We start by restating what the question is asking in probability notation.
The statement “half of the scholarship athletes are left-handers” translates to a conditional probability: the probability of being a left-hander given that the student is a scholarship athlete.
Now, evaluate each option:
Option 1: P( athlete | left-hander ). This reads as the probability that someone is an athlete given that they are a left-hander. It reverses the conditioning from the statement and does not match the form described in the problem.
Option 2: P( athlete and left-hander ). This is the joint probability that a student is both an athlete and a left-hander. While it relates to both properties, it does not express the conditional statement “half of the scholarship athletes are left-handers.”
Option 3: P( left-hander | athlete ). This is the conditional probability of being a left-hander given that the student is a scholarship athlete, which exactly corresponds to the phrase “half of the scholarship athletes are left-handers.”
Option 4: P( athletes or left-hander ). This denotes the probability of either being an athlete or being a left-hander (an improbable interpretation in this context) and does not represent the conditional relationship described.
Throughout these analyses, note that conditional notation is used to express a proportion within a specified subgroup (scholarship athletes), which is what the statement conveys. The other forms either reverse the conditioning, mix joint probability with a misinterpreted operation, or describe an unrelated union event.

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