CONDITIONAL PROBABILITY/ MUTUALLY EXCLUSIVE EVENTS

Question

The number of male and female students in section A whose favorite milk tea flavor is either chocolate or cookies and cream is shown below:

Chocolate:

Male: 8

Female: 12

Total: 20

Cookies and Cream:

Male: 10

Female: 10

Total: 20

*

Total of all male: 18

Total of all female: 22

Total of all students: 40

Questions:

1. What is the probability that a student is a male whose favorite flavor is chocolate or a female student whose favorite flavor is cookies and cream? Show the solution.

2. Find the probability that you choose a student who is male and his favorite flavor is cookies and cream then choose another student who is female and her favorite flavor is chocolate. Show the solution.

Answer 1

Often time with these questions a visualization is great! I tried to insert a drawing into this but unfortunately, it is too big, so I recommend drawing a two-way table.

1. What is the probability that a student is a male whose favorite flavor is chocolate or a female student whose favorite flavor is cookies and cream?

To calculate a probability like this, I like to think of it as the number of people that satisfy our conditions. There are 8 males who like chocolate, and there are 10 females who like Cookies and Cream. **Before you combine these together, ask yourself "Am I overcounting?"

In this situation no, since Male & Female are mutually exclusive, and favoritism is mutually exclusive as well. Therefore, there are 18 people who satisfy our condition and 40 total students. The probability is 18/40.

2. Find the probability that you choose a student who is male and his favorite flavor is cookies and cream then choose another student who is female and her favorite flavor is chocolate. Show the solution.

For questions like these, we have to understand dependence. The probability that we select a female after we select a male will change because we will be sampling without replacement. Here is the equation for the probability of two events that are dependent.

P( A & B) = P(A)P(B|A) or P(B)P(A|B). Sometimes one way is easier than the other. (As an exercise could you show these two are equal?)

P( Male likes Cookies and Cream & Female likes Chocolate ) = P(Male Likes Cookies and Cream)P(Female likes chocolate | male likes cookies and cream)

The probability that a male likes cookies and cream is 10/40 since in our table we see 10 students fall in that category and there are 40 students. After we select him there are 39 students. and 9 males that like cookies and cream. think of this as "updating your table."

Now- the probability of selecting a female that likes chocolate GIVEN we selected a male who likes cookies and cream is going to be 12/39.

Now we can multiply these two probabilities together.

(10/40)*(12/39) ≈ 0.0769

Hope this helps! Feel free to ask any questions if anything is unclear or doesn't make sense to you.