A card is drawn at random from a standard deck of playing cards. You win $4.00 if you draw a four or $25.50 if you draw an queen. Otherwise you lose $2. What is the expected value of the game?

Question

anonymous · Answer

Well, you need to first know what cards there are; queen, king, joker, spades... etc..

anonymous · Answer

there are four fours in a standard deck of 52 and four queens in a standard deck of 52

zepdrix · Answer

I'm not very good with probability, but lemme see if I can help. :)
So the expected value will be the sum of all the different possible outcomes, telling us how much we can expect to lose or gain from a single card draw.

In this case if a 4 is drawn we win $4. That will happen 4 out of 52 times. We can write it like this,$$\large $4\left(\frac{4}{52}\right)$$
If a queen is drawn, we win  $25.50. There is also a 4 in 52 chance of pulling a queen.$$\large $25.50\left(\frac{4}{52}\right)$$
We lose $2 on anything else. The way we write that as a fraction is - we take the REMAINING cards that we haven't considered and divide them by 52.
52-(4+4)=44
$$\large -$2\left(\frac{44}{52}\right)$$Understand why I wrote the negative?
This gives us an equation for expected value of,$$EV=\large $4\left(\frac{4}{52}\right) +\large $25.50\left(\frac{4}{52}\right)\large -$2\left(\frac{44}{52}\right)$$

zepdrix · Answer

Does that look right? I dunno what I'm doing :) Just trying to help lol.
I don't want to throw you off course if I'm not doing it correctly though XD

So if you calculate that, you get a positive EV of like 58 cents. So it would be profitable to play, since you can expect to make around 50 cents each draw.

anonymous · Answer

well that's exactly what i did. haha no joke. I'm 95% positive it's right haha. thank you so much, though!

zepdrix · Answer

Oh cool c: hah