Giving a test to a group of students, the grades and gender are summarized below

Question

A B C

Total Male 13 7 15 35

Female 19 2 20 41

Total 32 9 35 76

If one student is chosen at random, Find the probability that the student got an A:

Find the probability that the student was male AND got a "B":

Find the probability that the student was male OR got an "B":

If one student is chosen at random, find the probability that the student was male GIVEN they got a 'B':

Answer 1

Using the same set of data that you provided, I answered some similar questions and provided explanations that should help you answer the exact questions you were asked.

Question 1: What is the probability that a student earned a "B"?

Since there are no other conditions or requirements, we would just look at the total number of students who earned a "B" (9) out of the total number of students (76).

So P(earning a "B") = 9/76 = .1184 (approximately)

This should help you with the first question (What is the probability a student earned a "C"?)

Question 2: What is the probability that a student was female AND earned a "C"?

The word "and" indicates that we are looking for individuals that meet BOTH requirements mentioned. So we are going to look at the number of individuals who fall not just into one of these categories but both (20) out of the total number of students (76)

So P(female AND earned a "C") = 20/76 = .2632 (approximately)

This should help you with the second question (What is the probability that a student was male AND earned a "B"?)

Question 3: What is the probability that a student was female OR earned a "C"?

Unlike the word "and", the word "or" indicates that we are looking at individuals who met at least one of the requirements mentioned (in other words they could meet both, but they also could just be female or just have earned a "C" rather than doing both).

There are two different approaches (they are equally correct so go with what makes the most sense to you).

Option 1:

Look at all of the individual values (ignoring the total rows/columns) that represent people who met at least one of those requirements. In our data that would be individuals who were female & earned an A (19), female & earned a B (2), female & earned a C (20), male & earned a C (15) out of the total number of individuals (76). The key is making sure you do not double count the individuals who were female and earned a C.

So, P(female OR earned a "C") = (19 + 2 + 20 + 15)/76 = 56/76 = .7368 (approximately)

Option 2:

Look at how many total individuals were female (41) and the total who earned a "C" (35). A lot of people make the mistake of simply adding these two numbers together, but the problem with that is that both the 41 and the 35 include the individuals who met both requirements and we only want to include them once NOT twice. So, to avoid double counting those individuals, we add the totals mentioned together but then subtract out those who met both requirements( (that way they are no longer included twice, only once). It would look like this:

P(female OR earned a "C) = P(female) + P(earned a "C") - P(female AND earned a "C" = 41/76 + 35/76 - 20/76 = 56/76 = .7368 (approximately)

Using either of the two methods shown above will get the correct answer.

This should help with the third question (What is the probability that a student was male OR earned a "B"?)

Question 4: What is the probability that a student was female GIVEN that they earned a "C"?

When a condition is placed on the probability like this (the word GIVEN in this case), it indicates that we are no longer going to be looking at the overall total number of students but are limiting ourselves only to that given condition. In other words, since we are saying "given they earned a "C" this means that the denominator of the fraction we set up should only represent students who earned a "C" (so 35 not 76). And, out of those 35 students who earned a "C" how many were female (20... NOT 41... if you say 41 then you are including individuals who were no in the 35 who earned a C).

P(female GIVEN they earned a "C") = 20/35 = .5714 (approximately)

Note: It would be different if the question had asked "What is the probability that a student earned a "C" GIVEN that they were female. If that were the case the denominator would represent the female students. It would look like this:

P(earned a "C GIVEN female) = 20/41 = .4878 (approximately)

This should help with the last question (What is the probability that a student was male GIVEN that they earned a "B"?)