What is the expected number of boxes of cereals that he should buy?












1












$begingroup$



A company puts five different types of prizes into their cereal boxes,
one in each box and in equal proportions. If a customer decides to
collect all five prizes, what is the expected number of boxes of
cereals that he or she should buy?




TRY



Let $X$ be the number of boxes customer buys. For $i=1,2,3,4,5$, write



$$ X_{ij} = begin{cases} 1, & text{ith prize is inside jth box} \ 0, & text{otherwise} end{cases} $$



As I understand the problem, the number of boxes is not given so we may write



$$ X = sum_{j=1}^{infty} sum_{i=1}^5 X_{ij} $$



So



$$ E(X) = sum_{j geq 1 } sum_{i=1}^5 E(X_{ij}) $$



We know $E(X_{ij}) = P(X_{ij}=1)$ so we need to find proobability that ith prize is inside jth box. Here is the part where I get stuck. Am I appraoching this problem the correct way?










share|cite|improve this question









$endgroup$

















    1












    $begingroup$



    A company puts five different types of prizes into their cereal boxes,
    one in each box and in equal proportions. If a customer decides to
    collect all five prizes, what is the expected number of boxes of
    cereals that he or she should buy?




    TRY



    Let $X$ be the number of boxes customer buys. For $i=1,2,3,4,5$, write



    $$ X_{ij} = begin{cases} 1, & text{ith prize is inside jth box} \ 0, & text{otherwise} end{cases} $$



    As I understand the problem, the number of boxes is not given so we may write



    $$ X = sum_{j=1}^{infty} sum_{i=1}^5 X_{ij} $$



    So



    $$ E(X) = sum_{j geq 1 } sum_{i=1}^5 E(X_{ij}) $$



    We know $E(X_{ij}) = P(X_{ij}=1)$ so we need to find proobability that ith prize is inside jth box. Here is the part where I get stuck. Am I appraoching this problem the correct way?










    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$



      A company puts five different types of prizes into their cereal boxes,
      one in each box and in equal proportions. If a customer decides to
      collect all five prizes, what is the expected number of boxes of
      cereals that he or she should buy?




      TRY



      Let $X$ be the number of boxes customer buys. For $i=1,2,3,4,5$, write



      $$ X_{ij} = begin{cases} 1, & text{ith prize is inside jth box} \ 0, & text{otherwise} end{cases} $$



      As I understand the problem, the number of boxes is not given so we may write



      $$ X = sum_{j=1}^{infty} sum_{i=1}^5 X_{ij} $$



      So



      $$ E(X) = sum_{j geq 1 } sum_{i=1}^5 E(X_{ij}) $$



      We know $E(X_{ij}) = P(X_{ij}=1)$ so we need to find proobability that ith prize is inside jth box. Here is the part where I get stuck. Am I appraoching this problem the correct way?










      share|cite|improve this question









      $endgroup$





      A company puts five different types of prizes into their cereal boxes,
      one in each box and in equal proportions. If a customer decides to
      collect all five prizes, what is the expected number of boxes of
      cereals that he or she should buy?




      TRY



      Let $X$ be the number of boxes customer buys. For $i=1,2,3,4,5$, write



      $$ X_{ij} = begin{cases} 1, & text{ith prize is inside jth box} \ 0, & text{otherwise} end{cases} $$



      As I understand the problem, the number of boxes is not given so we may write



      $$ X = sum_{j=1}^{infty} sum_{i=1}^5 X_{ij} $$



      So



      $$ E(X) = sum_{j geq 1 } sum_{i=1}^5 E(X_{ij}) $$



      We know $E(X_{ij}) = P(X_{ij}=1)$ so we need to find proobability that ith prize is inside jth box. Here is the part where I get stuck. Am I appraoching this problem the correct way?







      probability






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      asked Dec 15 '18 at 6:44









      Jimmy SabaterJimmy Sabater

      2,161219




      2,161219






















          1 Answer
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          $begingroup$

          HINT: Consider instead: if I have found a given number of unique prizes, what is the probability that the next box I open has a prize I don't have yet? What is the expected time to get a new prize?



          SPOILER: this is the Coupon Collector's Problem.






          share|cite|improve this answer









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            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2












            $begingroup$

            HINT: Consider instead: if I have found a given number of unique prizes, what is the probability that the next box I open has a prize I don't have yet? What is the expected time to get a new prize?



            SPOILER: this is the Coupon Collector's Problem.






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              HINT: Consider instead: if I have found a given number of unique prizes, what is the probability that the next box I open has a prize I don't have yet? What is the expected time to get a new prize?



              SPOILER: this is the Coupon Collector's Problem.






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                HINT: Consider instead: if I have found a given number of unique prizes, what is the probability that the next box I open has a prize I don't have yet? What is the expected time to get a new prize?



                SPOILER: this is the Coupon Collector's Problem.






                share|cite|improve this answer









                $endgroup$



                HINT: Consider instead: if I have found a given number of unique prizes, what is the probability that the next box I open has a prize I don't have yet? What is the expected time to get a new prize?



                SPOILER: this is the Coupon Collector's Problem.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 15 '18 at 6:47









                Dan UznanskiDan Uznanski

                6,57021427




                6,57021427






























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