Question

it costs $2 to draw one card from a deck. if it is a face card, you win $7. if it is a seven you win $12. what is the expected value of this game?

Answer 1

Keep in mind the face cards. There are 52 cards in a deck, containing J's K's Q's. So, to figure out how many sets there are, divide by the 4 suits.

52/4= 13.

So now, face cards are 3/13 chance.
Add your 1/13 of getting a 7.

Now, you have a 4/13 chance of making your money. But the other 9/13? Bye bye $2.

That being said, now you have:
9/13 of losing $2
1/13 of gaining $10 [the profit]
3/14 of gaining $5 [profit again]

Now, my silly method at that point. Write 13 numbers down and then divide the result by 13.

10+5+5+5-2-2-2-2-2-2-2-2-2

So, -2 x 9 = -18 + 10 = -8 + 15= 7.

So, your expected profit is 7/13 of a dollar. Profit anyway, meaning youd make your initial $2 back. Making the expected value of this game approximately $2.54.