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Math Help - Two study questions

  1. #1
    Senior Member Danneedshelp's Avatar
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    Two study questions

    Q1: An urn contains 3 black, 4 red, and 5 white balls. Three balls are randomly drawn one at a time without replacement.
    a. Find the probability all three balls are of the same color.
    b. How many balls would you expect to be black or white?

    A1: a) Let N=12, n=3, r=3(same color balls)

    So, I used the above info to compute P(Y=3) using the hypergeometric dist and got 0.004545.

    b) I am a little thrown off by the "or" in the question. Some help would be great.

    Q2: Suppose Y , the number of fatal car accidents in a certain state, obeys a Poisson distribution with an average of four fatal accidents per day.
    a. For a particular day, what is the probability that there is at least one fatal car accident?
    b. Using Tchebysheff ís thoerem, find an upper bound for P (Y
    ≥ 12).

    A2: a) simple

    b) I am stuck on this part. Here is some of work...

    \mu=\lambda=4 and \sigma=2

    Given P(Y\geq\\12)\rightarrow by Tchebysheff's Theorem P(|Y-4|)\geq\\k2)\leq\\\frac{1}{k^{2}}. So, k=4 and thus P(|Y-4|)\geq\\8)\leq\\\frac{1}{16}

    That is what I have done so far, but I am not sure I am on the right track.

    Any help would be great. I am trying to straighten some things out before my exam on Thursday.

    Thanks
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Danneedshelp View Post
    Q1: An urn contains 3 black, 4 red, and 5 white balls. Three balls are randomly drawn one at a time without replacement.
    a. Find the probability all three balls are of the same color.
    b. How many balls would you expect to be black or white?

    A1: a) Let N=12, n=3, r=3(same color balls)

    So, I used the above info to compute P(Y=3) using the hypergeometric dist and got 0.004545.
    What is the probability of selecting three black balls? Three red balls? Three white balls?

    CB
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Danneedshelp View Post
    Q1: An urn contains 3 black, 4 red, and 5 white balls. Three balls are randomly drawn one at a time without replacement.
    a. Find the probability all three balls are of the same color.
    b. How many balls would you expect to be black or white?

    A1: a) Let N=12, n=3, r=3(same color balls)

    So, I used the above info to compute P(Y=3) using the hypergeometric dist and got 0.004545.

    b) I am a little thrown off by the "or" in the question. Some help would be great.
    There are now two types of ball 8 (black or white) balls and 4 red balls.

    So the expected number of (black or white) balls is:

    E(n)=0 \times p(0)+1 \times p(1) + 2 \times p(2) + 3 \times p(3)

    where p(r) is the probability of exactly r (black or white) balls in three draws without replacement.

    CB
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  4. #4
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    What is the probability of selecting three black balls? Three red balls? Three white balls?

    CB
    So, If I let B represent the event of choosing three black balls, R the event of choosing three red balls, and W represent the event of choosing 3 white balls I need to find

    P(B\cup\\R\cup\\W)=P(B)+P(R)+P(W) by using the hypergeometric dist for each probability. So, I only need to change my r value each time. Correct?
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  5. #5
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    There are now two types of ball 8 (black or white) balls and 4 red balls.

    So the expected number of (black or white) balls is:

    E(n)=0 \times p(0)+1 \times p(1) + 2 \times p(2) + 3 \times p(3)

    where p(r) is the probability of exactly r (black or white) balls in three draws without replacement.

    CB
    The r refers the collection of objects with the desired properties, so in this case, r=8(black or white). Hence, each probability you showed above is hypergeomentric with N=12, n=3, and r=8?

    Thanks a lot for your help, I really appreciate it.

    Can someone check my work for the Tchebysheffs propblem
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by Danneedshelp View Post
    So, If I let B represent the event of choosing three black balls, R the event of choosing three red balls, and W represent the event of choosing 3 white balls I need to find

    P(B\cup\\R\cup\\W)=P(B)+P(R)+P(W) by using the hypergeometric dist for each probability. So, I only need to change my r value each time. Correct?
    You should not need to use the hypergeometric distribution explicitly, you should just be able to write these down.

    Your P(B)=(3/12)(2/11)(1/10) etc.

    CB
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  7. #7
    Grand Panjandrum
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    Quote Originally Posted by Danneedshelp View Post
    The r refers the collection of objects with the desired properties, so in this case, r=8(black or white). Hence, each probability you showed above is hypergeomentric with N=12, n=3, and r=8?

    Thanks a lot for your help, I really appreciate it.

    Can someone check my work for the Tchebysheffs propblem
    No that is not what I wrote, and I did mean exactly what I did write.

    CB
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