What is expected number of cards drawn when a card is chosen from a 52 deck and game ends on picking a heart? I need help going about solving this problem:
Suppose a card game is played using a standard 52 card deck.  Drawing a heart ends the game.  Drawing anything other than a heart continues the game.  There is no replacement of the drawn card back into the deck.
What is the expected number of cards drawn before the game ends (i.e when a heart is chosen)?

For the 39 non-hearts, define $A_i,\, 1\leq i\leq 39 $ to be the event that card $i$
appears before all the hearts. The number of draws until we get the first heart is
$X=1+\sum_{i=1}^{39} 1_{A_i}$, and since $\mathbb{P}(A_i)=1/14$ we get
$$\mbox{average number of draws}
=\mathbb{E}(X)=1+\sum_i \mathbb{P}(A_i)=53/14\approx 3.79.$$
