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Math Help - Why do I get a different answer using the binomial distribution function?

  1. #1
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    Why do I get a different answer using the binomial distribution function?

    The question I am currently working with is as follows:

    If three cards are randomly selected from a standard 52 card deck of playing cards, what is the probability that at least one of them will be a painted face card (king, queen or jack)?

    Initially, I just went ahead and solved it as follows:

    P(at least 1 will be a face card) = 1 - (40/52)*(39/51)*(38/50) = 55.29%

    However, I went to solve it using the binomial distribution function just out of curiosity and got a slightly different answer:

    P(at least 1 will be a face card) = 1 - (3 choose 3) * (40/52)^3 * (12/52)^0 = 54.48%

    Why?
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  2. #2
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    Re: Why do I get a different answer using the binomial distribution function?

    Quote Originally Posted by terrorsquid View Post
    The question I am currently working with is as follows: If three cards are randomly selected from a standard 52 card deck of playing cards, what is the probability that at least one of them will be a painted face card (king, queen or jack)?
    Initially, I just went ahead and solved it as follows:

    P(at least 1 will be a face card) = 1 - (40/52)*(39/51)*(38/50) = 55.29%

    However, I went to solve it using the binomial distribution function just out of curiosity and got a slightly different answer: P(at least 1 will be a face card) = 1 - (3 choose 3) * (40/52)^3 * (12/52)^0 = 54.48%
    The binomial distribution cannot be use here. The cards are not independent events.
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  3. #3
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    Re: Why do I get a different answer using the binomial distribution function?

    Hello, terrorsquid!

    If three cards are randomly selected from a standard 52 card deck of cards,
    what is the probability that at least one will be a face card (K, Q or J)?

    Initially, I just went ahead and solved it as follows:

    P(at least 1 face card) = 1 - (40/52)*(39/51)*(38/50) = 55.29%

    However, I solved it using the binomial distribution function just out of curiosity
    and got a slightly different answer:

    P(at least 1 face card) = 1 - (3 choose 3) * (40/52)^3 * (12/52)^0 = 54.48%

    Why?

    Plato is correct . . . The draws are not independent.

    With the binomial function, you are using p = \tfrac{40}{52} .for each draw.
    . . You are drawing the 3 cards with replacement.


    Here's another approach . . .

    There are 12 Face cards and 40 Others.

    Select 3 cards from the 52 cards.
    . . There are: . {52\choose3} \:=\:22,\!100\text{ ways.}

    Select 3 cards from the 40 Others.
    . . There are: . {40\choose3} \:=\:9,\!880\text{ ways.}

    \text{Hence: }\:P(\text{no Face cards}) \;=\;\frac{9,\!880}{22,\!110} \:=\:\frac{38}{85}

    \text{Therefore: }\:P(\text{at least one Face card}) \;=\;1 - \frac{38}{85} \;=\;\frac{47}{85}
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    Re: Why do I get a different answer using the binomial distribution function?

    Thank you! Very helpful.

    I am starting a statistics degree next month and was interested in going over probability and some of the topics I will cover in my course. I was wondering if you could recommend any resources (perhaps a textbook) that do a good job of covering some of the major topics that I will come across - just something structured so I can progress through the various components of probability in a logical order.

    Thanks.
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    Re: Why do I get a different answer using the binomial distribution function?

    Quote Originally Posted by terrorsquid View Post
    I was wondering if you could recommend any resources (perhaps a textbook) that do a good job of covering some of the major topics that I will come across - just something structured so I can progress through the various components of probability in a logical order.
    Here is a great website.
    Some where there is a freepdf of a textbook, I think.
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