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Probabilities and Card Tricks

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Consider the experiment of drawing two cards from a deck and adding their values (with ace = 1).

a. Describe the outcomes of this experiment. List the elements of the sample space.
b. What is the probability of obtaining a total of 5 for the two cards?
c. Let A be the event "total card value is 5 or less." Find P(A) and P(Ac).

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As the solution walks through this problem, it gives a step-by-step explanation on how to approach this type of probability problem.

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a. If we label a card with its number and color (e.g. Heart Ace is written as 1H, Spade King is 13S, Diamond 10 is 10D, Club 4 is 4C), then each outcome is a pair of two distinct cards (e.g. (1H,2S), (3S,2C), etc.).

We can write down a few examples from this sample space:
(1H,2H),(1H,3H),....,(2H,3H),(2H,4H),...,...
(1H,2S),(1H,3S),....

Basically, the sample space can be represented as {(nX,mY), where n,m are numbers between 1 and 13, X,Y chosen from H,S,D,C, and nX is different from mY, and we don't count order, meaning (nX,mY) is the same as (mY,nX)}.

b. Since there is a total of 52 cards, there are 52 choices. 2 = 52*51/2=1326 choices of two cards.

Let's calculate how many of ...

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