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Math Help - Collecting coupons

  1. #1
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    Collecting coupons

    Suppose that there are N distinct tyoes of coupons and each time one obtains a coupon it is, independent of prior selections, equally likely to be any one of the N types. One random variable of interest is T, the number of coupons that needs to be collected until one obtains a complete set of at least one of each type. Rather than derive P\{T=n\} directly, let us start by considering the probability that T is greater than n. To do so, fix n and define the events A_{1}, A_{2}, ..., A_{N} as follows: A_{j} is the event that no type j coupon is contained among the first n, j=1, ..., N.
    Hence, P\{T>n\}=P\left(\bigcup^{N}_{j=1}A_{j}\right)

    I coulnd't understand the last equality, and why can't we derive P\{T=n\} directly?

    Appreciate those who help!!
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  2. #2
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    Quote Originally Posted by acc100jt View Post
    Suppose that there are N distinct tyoes of coupons and each time one obtains a coupon it is, independent of prior selections, equally likely to be any one of the N types. One random variable of interest is T, the number of coupons that needs to be collected until one obtains a complete set of at least one of each type. Rather than derive P\{T=n\} directly, let us start by considering the probability that T is greater than n. To do so, fix n and define the events A_{1}, A_{2}, ..., A_{N} as follows: A_{j} is the event that no type j coupon is contained among the first n, j=1, ..., N.
    Hence, P\{T>n\}=P\left(\bigcup^{N}_{j=1}A_{j}\right)

    I coulnd't understand the last equality, and why can't we derive P\{T=n\} directly?

    Appreciate those who help!!
    The equality is just a translation from words to maths of a simple statement, namely "The event \{T>n\} means that among the first n coupons there is a missing type j\in\{1,\ldots,N\} of coupons". This should be clear (otherwise, give it a second thought ; remember T is the first time we have all types of coupons).
    Then \bigcup_{j=1}^N A_j is the event "There exists j\in\{1,\ldots,N\} such that A_j happens", and here we are since A_j is defined as: "the type j is missing among the n first coupons".

    About P(T=n), if you can derive it directly, that's just fine! It is probably easier however (here and in a large variety of situations) to find P(T>n) and then deduce the first one by P(T=n)=P(T>n-1)-P(T>n).
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