Math question?

Question

40% of students are over 25. If 5 students are randomly selected, what is the probability that exactly 2 will be over 25?

a. .3456

b. .2212

c. .3975

d. .3633

How did you get this answer?

Answer 1

That's a binomial experiment with p = .4, n=5, and k = 2. P is your probability of "success" (not necessarliy sometihng good but the result you're interested in), n is your number of trials (5 students = 5 trials) and k is your number of "successes" (2 students over 25.
Work out C(5,2)x .4^2 x .6^3 and you get
10 x ,16 x .216 = .3456 (a)
To understand why you use combinations to work out binomial probabilities, try drawing a tree diagram for a simpler binomial experiment, such as one with n = 2 and p = .4. You'll see where the combinations come in, especially if you extend it out to another level, n = 3. Much beyond that and tree diagrams get unweildy, but they are very helpful in seeing how, in the middle part, you wind up using combinations.

Answer 2

This is a binomial model with p=0.40

You want 5C2p^2(1-p)^3

Do you really need further explanation?

Answer 3

There are 5C2 ways for this to happen, and the probability for each is 0.4*0.4*0.6*0.6*0.6, so
p(exactly 2) = 5C2(0.4)^2(0.6)^3