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Math Help - probability problems

  1. #1
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    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. #2
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    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: . {52\choose3} \,=\,22,\!100 possible outcomes.

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


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




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

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

    . . . {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 r heads occurring is given by the term containing p^r
    in the binomial expansion of (p+q)^{20}, where p =q = \tfrac{1}{2}

    Calculate the probability that exactly 9 heads will be tossed.

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

    We want the term with 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}: . {5\choose2}{4\choose2}{3\choose3} \:=\:10\cdot6\cdot1 \:=\:60 ways.

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

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


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

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