Question

You are dealt one card from a full deck of 52 cards and your opponent is dealt two cards (without replacement). If you get a card between 6 and 10, your opponent pays you 4, and if you get a King or Queen, your opponent pays you 3. If you don't have a 6-10, Queen or King, but you have more hearts than your opponent, your opponent pays you 1. In all other cases you pay 2. What is the expectation of your winning? Take you paying to be interpreted as negative winning and you receiving as positive winning.

Answer 1

The probability that I'll get a card between 6 and 10 is given by:

One card from the deck of 52 cards can be drawn in 52 ways.
There are 5 cards between 6 and 10 for each suit and there are 4 suits in a deck.
So, One card between 6 and 10 can be drawn in 20 ways.
So, the probability that I'll get a card between 6 and 10 is given by:

The probability that I'll get a king is given by:
One card from the deck of 52 cards can be drawn in 52 ways.
There are 4 'king' cards and 4 'queen' in the deck, each of the 4 suit has one 'king' card & one 'queen' card.
So, one 'king\queen' card can be chosen from the 8 'king\queen' cards in 8 ways.
So, the probability that I'll get a king is given by:


The probability that I will get more hearts than my opponent:
I can get only one heat at most.
So, to win, my opponent cannot pick up a heart.
One card from the deck of 52 cards can be drawn in 52 ways.
There are 13 'hearts' in a deck. So one 'heart' can be chosen from these 13 cards in 13 ways.
Probability that, I can get only one heat at most is : .

As I have already taken out one card, my opponent is left with 51 cards.
My opponent can pick two cards from 51 cards in : ways.

My opponent can not pick any of the remaining 'hearts'. Therefore he has only (52-1-12) = 39 choices..

So,My opponent can pick 2 cards from these 39 cards in ways.

The probability that My opponent will not get any heart =

Since these draws are independent, the probability that I will get more hearts than my opponent:

.


Let, X be a random variable denoting my pay. X is structured as:

According to the given scenario, X has probability distribution as follows:


My expectation of winning:

E(X)= (4*0.3846)+(3*0.1538)+(1*0.145275)+(-2*0.316325)
=1.512425

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