Probability

[Starling]

Hello!

I have a 2 part probability problem and I'm hoping I could get some help.

The problem is...

Pick 4 cards from 52; you win if all four cards are hearts. Pr(win)=?
*I got 13/52 ...but I think it may be wrong since it was too easy to answer :?

The 2nd part is...

If you win the card-draw game, then you are paid $1000; if you lose, you pay $3. What is the expected value, to you, of this bet?
*My teacher never went over expected value so I am somewhat lost over the problem. Any help would be great!

 

[wjm11]

Pick 4 cards from 52; you win if all four cards are hearts. Pr(win)=?

First card pick P = 13/52 = ¼
Second card pick, P = 12/51 = 4/17
(Both the number of hearts and number of cards remaining in the deck are reduced by one.)
Third card pick, P = 11/50
Fourth card pick, P = 10/49
Multiply these probabilities:

(1/4)(4/17)(11/50)(10/49) = .00264

 

[soroban]

Hello, Starling!

Are you familiar with "Combinations"?

Pick 4 cards from 52.
You win if all four cards are hearts.
Pr(win) = ?

There are 52 cards; we will pick 4 of them.
\(\displaystyle \;\;\)There are: \(\displaystyle \,C(52,4)\;=\;\frac{52!}{4!\cdot48!}\;=\;270,725\) ways.

There are 13 hearts; we want 4 of them.
\(\displaystyle \;\;\)There are: \(\displaystyle \,C(13,4)\;=\;\frac{13!}{4!\cdot9!}\;=\;715\) ways.

Therefore: \(\displaystyle P(\text{4 hearts})\;=\;\frac{715}{270,725}\;=\;\frac{11}{4165}\)

If you win the card-draw game, then you are paid $1000.
I you lose, you pay $3.
What is the expected value, to you, of this bet?

\(\displaystyle \frac{11}{4165}\) of the time, you will win $1000.

The other \(\displaystyle \frac{4154}{4165}\) of the time, you will lose $3.

\(\displaystyle \;\;E\;=\;\left(\frac{11}{4165}\right)(1000)\,+\,\left(\frac{4154}{4165}\right)(-3)\;=\;\frac{-1462}{4165}\;=\;-0.351020408\)

Therefore, you can expect to lose an average of 35 cents per game.


Edit: Corrected the typo . . . Thanks, Galactus!

 

[wjm11]

Thanks, Soroban. Minor typo, though: (11/4165) is multiplied by 1000 in the final eqn., not 3. The answer is correct however.

 

[Starling]

You guys are unbelieveably awesome...great step by step directions! Thanks so much for the help!