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Math Help - Hypergeometric Distribution

  1. #1
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    Hypergeometric Distribution

    I would greatly appreciate if someone could help me set up and solve the following two problems.

    1. If 7 cards are dealt from an ordinary deck of 52 cards, what is the probability that exactly 2 of them will be face cards? That at least 1 of them will be a queen?

    Using the hypergeometric distribution formula, I said that this would be equal to 12 choose 3 divided by 52 choose 7. What am I doing wrong?

    2. A random committee of size 3 is selected from 4 doctors and 2 nurses. Write a formula for the probability distribution of the random variable X representing the number of doctors on the committee. Find P(2<=X<=3).

    Thanks!
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  2. #2
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    Hello, Walcott89!

    1. 7 cards are dealt from an ordinary deck of 52 cards.
    (a) What is the probability that exactly 2 of them will be face cards?
    There are: . {52\choose7} possible outcomes.


    There are: .12 Face Cards and 40 Others.

    We want 2 Face Cards and 5 Others.
    . . There are: . {12\choose2}{40\choose5} ways.

    Therefore: . P(\text{exactly 2 Face Cards}) \;=\;\dfrac{{12\choose2}{40\choose5}}{{52\choose7}  }




    [(b) What is the probability that at least 1 of them will be a Queen?
    There are: 4 Queens and 48 Others.

    Find the probability of no Queens.
    Then we want 7 Others.
    . . There are: . {48\choose7} ways.
    Hence: . P(\text{no Queens}) \;=\;\frac{{48\choose7}}{{52\choose7}}
    Therefore: . P(\text{at least one Queen}) \;\;=\;\;1 - \frac{{48\choose7}}{{52\choose7}}


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  3. #3
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    Quote Originally Posted by Walcott89 View Post
    I would greatly appreciate if someone could help me set up and solve the following two problems.

    1. If 7 cards are dealt from an ordinary deck of 52 cards, what is the probability that exactly 2 of them will be face cards? That at least 1 of them will be a queen?

    Using the hypergeometric distribution formula, I said that this would be equal to 12 choose 3 divided by 52 choose 7. What am I doing wrong?

    2. A random committee of size 3 is selected from 4 doctors and 2 nurses. Write a formula for the probability distribution of the random variable X representing the number of doctors on the committee. Find P(2<=X<=3).

    Thanks!

    2.

    you have to choose x doctors and (3-x) nurses in the sample of 3

    C(3,x) C[2,(3-x)] / C(6,3)


    binomial distributionmixture: Binomial distribution
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