Probability (Counting Methods)

Question

A math class consists of 34 students, 12 female and 22 male. Three students are selected at random, one at a time, to participate in a probability experiment (selected in order without replacement).

(a) What is the probability that a male is selected, then two females?

(b) What is the probability that a female is selected, then two males?

(c) What is the probability that two females are selected, then one male?

(d) What is the probability that three males are selected?

(e) What is the probability that three females are selected?

Answer 1

Probability is all about counting a sample of something. So we have a sample of 34 people and two smaller samples of 22 males and 12 females within that sample. We are going to select people from these samples based on what each problem tells us, and as we do so we need to reduce the count of our samples:

a.) Select 1 male then 1 female then 1 female:

Select 1 male: there are 22 males and 34 people, so the probability of selecting one male is 22/34.

Then select 1 female: there are now 33 total people because we have selected 1 male above. So now

we have 12 females and 33 people in our sample, and the probability of

selecting 1 female is then 12/33.

Then select 1 female: there are now 32 total people remaining in our sample because we have

selected 1 male and 1 female above. Similarly, there are now 11 females

remaining because we selected 1 female above. The probability of selecting a

female now is 11/32.

Now we have to decide if we want to select

"1 male OR or 1 female OR 1 female",

or do we want to select

"1 male AND 1 female AND 1 female".

This is important because the "OR" means that we ADD all of the probabilities above together, while the "AND" means that we MULTIPLY all of the probabilities together. From reading the problem we want to select 1 male AND 1 female AND then another female, so we will multiply all of our probabilities above together to get our answer:

22/34 * 12/33 * 11/32 = 11/136

b.) Following the process above, we want to select 1 female AND 1 male AND then another male. So we find our probabilities and MULTIPLY them:

12/34 * 22/33 * 21/32 = 21/136

c.) Select 1 female AND then another female AND then 1 male.

12/34 * 11/33 * 22/32 = 11/136

d.) Select 1 male AND then 1 male AND then 1 male.

22/34 * 21/33 * 20/32 = 35/136

e.) Select 1 female AND 1 female AND 1 female.

12/34 * 11/33 * 10/32 = 5/136