Geometric distribution. Johnny has 1,176 Pok´emon cards in total. Pok´emon EX is a special type of card, and Johhny has 39 EX-type cards. He is looking for an EX-type card, but all of the cards are completely mixed up and stored in a shoe box. His mother is calling him for dinner. What is the probability that Johnny will have to look through no more than 25 cards before he finds an EX-type card?

Question

Agour2q9 · Accepted Answer

Distribution: For   x  =  x   these must be a run of   (  x  −  1  )   failure followed by a success, so    P  (  x  =  x  )  =  θ  (  1  −  θ      )          x      −      1          x  =  1  ,  2  ,  3  ,  .  .  .  (  0  &amp;lt;  θ  &amp;lt;  1  )    I knows way of the geometric distribution let x be the numbers of failure&#039;s between the first success    than  P  (  x  =  x  )  =  θ  (  1  −  θ      )    x      x  =  0  ,  1  ,  2  ,  3  ,  .  .  .    P  (  x  ≤  25  )    =  P  (  x  =  0  )  +  P  (  x  =  1  )  +  −  −  +  P  (  x  =  24  )    +  P  (  x  =  25  )    =  θ  (  1  −  θ      )    0    +  θ  (  1  −  θ      )    1    +  (  1  −  θ      )    2      +  −  −  +  θ  (  1  −  θ      )          25          =  θ  [  1  +  (  1  −  θ  )  +  (  1  θ      )    2    +  −  −  +  (  1  −  θ      )          25        ]    =  θ  ∗            [      1      −      (      1      −      θ              )                  26                    ]              [      1      −      (      1      −      θ      )      ]          =      θ    θ    [  1  −  (  1  −  θ      )          26        ]  =  1  −  (  1  −  θ      )          26          where   θ  =      39    1176      P  (  x  ≤  25  )  =  1  −  [  1  −      39    1176        ]          26          1  −            39      1176              26          =  0.5839