Question

3. Cards: I take a standard deck of 52 cards, consisting of 13 spades, 13 hearts, 13 diamonds, and 13 clubs. I am interesting in seeing how many non-heart cards I can draw before picking a heart. After each draw, I will put the card back in the deck, so there is a 1/4 chance I get a heart with each draw, and a 3/4 chance I do not get a heart. I think about this a ittle bi and come up with the following probability model for X, the number of non-hearts drawn before I get a heart. 72 n=0, 1, 2, . .. (a) Using the fact that the probability space must have a total measure of one, find the value of c. (Yes, I know there is another way to find c directly from the experiment description, but I want you to use/remember this important infinite series. Of course, the answer you get should be consistent with what you would expect from the experiment.) (b) Find P(X22). (c) Find P(X є {0.2, 4, 6, . . .)).

Answer 1

(b) P(x>=2) = 1 - P(x = 0) - P(x = 1) - P(x = 2)

   P(x = n) = ⁵²C_n (3/4)ⁿ(1/4)^52-n

    P(x = 0) = ⁵²C₀ (3/4)⁰(1/4)^52-0

                     = (1/4)⁵²

   P(x = 1) = ⁵²C₁(3/4)¹(1/4)^52-1

                   = 156(1/4)⁵²

     P(x = 2) = ⁵²C₂ (3/4)²(1/4)^52-2

                            =11934(1/4)⁵²

      P(x>=2) = 1 - [ 12091 ](1/4)⁵²