# Thread: probability problems

1. ## probability problems

I need just a little help answering some math questions. Maybe someone could help?

1.) What is the probability of drawing EXACTLY 2 red cards in a hand of 3 cards frawn from a deck of 52 cards?

2.) State the coefficient of the term containing X^-11 in the expansion of (4X^2+X^-3)^7?

3.) A coin is tossed 20 times in a row. The probability of exactly r heads occurring is given by the term containing p^r in the binomial expansion of (p+q)^20, where p=q=0.5. Calculate the probability that exactly 9 heads will be tossed.

4.) A bag contains 12 blocks, 5 of which are red, 4 are blue, and the rest are green. If 7 blocks are selected randomly, determine the number of ways that AT LEAST 2 blocks of each color are included.

5.) A group of vehicles consists of 7 [COLOR=green! important]cars
and 6 trucks. Five are chosen randonly to be in the show room at a car [COLOR=green! important]dealership[/COLOR]. Determine the probability that AT LEAST one of each vehicle is chosen.

6.) The mathematics department has five committees. Each of these committees meets once a month. Membership on these committees is as follows.

Committee A: Shawn, Larry, Elliott
Committee B: Elliott, Warner, Henry, Kelly
Committee C: Warner, Larry
Committee D: Andrew, Larry, Shawn
Committee E: Benny, Candy, Kelly, Shawn

What is the minimum number of time slots needed to schedule the committee meetings with no conflicts?

So I these are just the questions I need answered. Hope someone can help. I think I got number 1 and number 5 but I just want to make sure. Thanks.
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2. Hello, Jon!

Here are a few of them . . .

1) What is the probability of drawing exactly 2 red cards
in a hand of 3 cards drawn from a deck of 52 cards?

There are: .$\displaystyle {52\choose3} \,=\,22,\!100$ possible outcomes.

We want two Reds and one Black.
. . There are: .$\displaystyle {26\choose2}{26\choose1} \:=\:325\cdot26 \:=\:8,\!450$ ways.

Therefore: .$\displaystyle P(\text{2 Reds}) \:=\:\frac{6,\!450}{22,\!100} \:=\:\frac{13}{34}$

2) State the coefficient of the term containing $\displaystyle x^{-11}$ in the expansion of $\displaystyle \left(4x^2+x^{-3}\right)^7$

In the expansion of $\displaystyle \left(4x^2+x^{-3}\right)^7$, the only term with $\displaystyle x^{-11}$ is:

. . . $\displaystyle {7\choose2}\left(4x^2\right)^2\left(x^{-3}\right)^5 \;=\;21\left(16x^4\right)\left(x^{-15}\right) \;=\;\boxed{336}\,x^{-11}$

3) A coin is tossed 20 times in a row.
The probability of exactly $\displaystyle r$ heads occurring is given by the term containing $\displaystyle p^r$
in the binomial expansion of $\displaystyle (p+q)^{20}$, where $\displaystyle p =q = \tfrac{1}{2}$

Calculate the probability that exactly 9 heads will be tossed.

We have: .$\displaystyle (p + q)^{20} \:=\:\left(\tfrac{1}{2} + \tfrac{1}{2}\right)^{20}$

We want the term with $\displaystyle p^9\!: \;\;{20\choose9}\left(\tfrac{1}{2}\right)^9\left(\ tfrac{1}{2}\right)^{11} \;=\; (167,\!960)\left(\frac{1}{1,\!048,\!576}\right) \;=\;\boxed{\frac{20,\!995}{131,\!072}}$

4) A bag contains 12 blocks: 5 red, 4 blue, and 3 green.
If 7 blocks are selected randomly, determine the number of ways
that at least 2 blocks of each color are drawn.
There are three cases . . .

. . (1) {RR|BB|GGG}: .$\displaystyle {5\choose2}{4\choose2}{3\choose3} \:=\:10\cdot6\cdot1 \:=\:60$ ways.

. . (2) {RR|BBB|GG}: .$\displaystyle {5\choose2}{4\choose3}{3\choose2} \:=\:10\cdot4\cdot3 \:=\:120$ ways.

. . (3) {RRR|BB|GG}: .$\displaystyle {5\choose3}{4\choose2}{3\choose2} \:=\:10\cdot6\cdot3 \:=\:180$ ways.

Therefore, there are: .$\displaystyle 60 + 120 + 180 \:=\:\boxed{360}$ ways.