Answer to Question #104529 in Statistics and Probability for Joseph

Question

3. Our class is creating a new academic team , the data managers. The team will have 4 members to perform different tasks: collector , analyzer, grapher , and recorder . 10 people in class want to
be on the team, so we pick a random team out of 10.

a) if exactly 4 boys volunteer, what is probability that the team is all boys or no boys ? Evaluate your answer

b) what is probability that student 1 and student 2 are on the same team? Evaluate your answer


c) with two teacher coaches, the team of four takes a picture (6 people in picture ) what is the probability that the teachers are together in the picture? Evaluate your answer .

Answer 1

The possible number of ways for creating a new academic team is:

$$\binom{10}{4}={10! \over 4!(10-4)!}={10\cdot9\cdot8\cdot7 \over 1\cdot2\cdot3\cdot4}=210$$

a)

If exactly 4 boys volunteer, what is probability that the team is all boys ?

$$P(all \ boys )={\binom{4}{4}\binom{10-4}{4-4} \over\binom{10}{4}}={1\cdot1 \over 210}={1 \over 210}$$

If exactly 4 boys volunteer, what is probability that the team is no boys?

$$P(no \ boys )={\binom{4}{0}\binom{10-4}{4-0} \over\binom{10}{4}}={1\cdot{6! \over 4!(6-4)!} \over 210}={15 \over 210}$$

If exactly 4 boys volunteer, what is probability that the team is all boys or no boys ?

$$P(all \ boys\ or\ no\ boys )={1 \over 210}+{15 \over 210}={16 \over 210}={8 \over 105}$$

b)

What is probability that student 1 and student 2 are on the same team?

$$P(student\ 1\ and\ student\ 2\ boys )={\binom{10-2}{4-2} \over\binom{10}{4}}=$$ $$={{8! \over 8!(8-2)!} \over 210}={{8\cdot7 \over 2} \over 210}={2 \over 15}$$

c) Consider two teacher coaches as unit.

With two teacher coaches, the team of four takes a picture (6 people in picture ) what is the probability that the teachers are together in the picture?

$$P(teachers\ are\ together )={2!\cdot5! \over 6!}={1 \over 3}$$