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Math Help - Need help in finding the probability of events occuring?

  1. #1
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    Need help in finding the probability of events occuring?

    Question:
    i have a total of 16 events whereby the probability of 2 events occuring is 0.3, the probability of 6 events occuring is 0.003 and the probability of 8 events happening is 0.0003.

    How can i find out the probability of 1 event occuring out of the 16 events?
    How can i find out the probability of 5 events occuring out of the 16 events?
    How can i find out the probability of 10 events occuring out of the 16 events?

    Assumptions:
    the 16 events occuring are considered to be independent events.

    Note:
    i cannot use binomial distribution to solve as there are 3 different probabilties for the 3 cases. is this question considered to be multinomial distribution?

    Thanks for your help
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  2. #2
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    Quote Originally Posted by spidermman View Post
    Question:
    i have a total of 16 events whereby the probability of 2 events occuring is 0.3, the probability of 6 events occuring is 0.003 and the probability of 8 events happening is 0.0003.

    How can i find out the probability of 1 event occuring out of the 16 events?
    How can i find out the probability of 5 events occuring out of the 16 events?
    How can i find out the probability of 10 events occuring out of the 16 events?

    Assumptions:
    the 16 events occuring are considered to be independent events.

    Note:
    i cannot use binomial distribution to solve as there are 3 different probabilties for the 3 cases. is this question considered to be multinomial distribution?

    Thanks for your help
    I don't think there's enough information. Is this the entire question?
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  3. #3
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    To make the question clearer, i meant that the 2 events have individual probability of 0.3 occuring, 6 events have individual probability of 0.003 occuring and 8 events have individual probability of 0.0003. Hence, there are a total of 16 individual events.

    Assume that the 16 events will occur and they are independent,
    i would like to find out that out of this 16 events, what is the probability that 5 events occured while 11 events does not occur?

    and i would also like to find out the probability of number (from 0 to 16) of events occuring?
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  4. #4
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    Quote Originally Posted by spidermman View Post
    To make the question clearer, i meant that the 2 events have individual probability of 0.3 occuring, 6 events have individual probability of 0.003 occuring and 8 events have individual probability of 0.0003. Hence, there are a total of 16 individual events.

    Assume that the 16 events will occur and they are independent,
    i would like to find out that out of this 16 events, what is the probability that 5 events occured while 11 events does not occur?

    and i would also like to find out the probability of number (from 0 to 16) of events occuring?
    This question still makes no sense to me. Not for the least reason being that (2)(0.3) + (6)(0.003) + (8)(0.0003) = 0.6204 so one of the 16 events is not certain to occur. And since the events have different probabilities it makes no sense to ask about the probability of a particular number of events occuring.

    Is this exactly how the question is stated?
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  5. #5
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    The question goes like this: Probability of occuring
    Event #1: 0.0003
    Event #2: 0.3
    Event #3: 0.003
    Event #4: 0.003
    Event #5: 0.0003
    Event #6: 0.0003
    Event #7: 0.003
    Event #8: 0.0003
    Event #9: 0.0003
    Event #10: 0.0003
    Event #11: 0.003
    Event #12: 0.003
    Event #13: 0.003
    Event #14: 0.0003
    Event #15: 0.3
    Event #16: 0.0003

    So lets call X{i} to be a bernoulli random variable that is 1 when Event #i happens. And Y = sum{Xi} for 1 to 16.

    The question is what (i) P (Y=5), (ii) P(Y=10), (iii) P(Y=15) ?
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  6. #6
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    Quote Originally Posted by spidermman View Post
    The question goes like this: Probability of occuring
    Event #1: 0.0003
    Event #2: 0.3
    Event #3: 0.003
    Event #4: 0.003
    Event #5: 0.0003
    Event #6: 0.0003
    Event #7: 0.003
    Event #8: 0.0003
    Event #9: 0.0003
    Event #10: 0.0003
    Event #11: 0.003
    Event #12: 0.003
    Event #13: 0.003
    Event #14: 0.0003
    Event #15: 0.3
    Event #16: 0.0003

    So lets call X{i} to be a bernoulli random variable that is 1 when Event #i happens. And Y = sum{Xi} for 1 to 16.

    The question is what (i) P (Y=5), (ii) P(Y=10), (iii) P(Y=15) ?
    Looks to me like \Pr(Y = 5) = \sum_{a,b,c} \frac{5!}{a! \, b! \, c!} (0.3)^a (0.003)^b (0.0003)^c

    where a + b + c = 5 subject to the restrictions 0 \leq a \leq 2, 0 \leq b \leq 6 and 0 \leq c \leq 8.

    etc.

    There might be a clever way of avoiding the tedium of working out the required combinations of a, b and c but I don't see it right now.
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