**What is probability?**

In order to understand probability, you must know how many possible ways a thing can happen. If I flip a coin, how may ways can it land? There are*two*possible ways. If we want to calculate the probability of the coin landing a head, we see that the head is one of two possible ways so the probability is 1/2 or .5.

Now, how many ways can a single die land? This is 6 because there are six faces of a die. What is the probability of rolling a 3 with a single die? 1/6 because there is one "3" and 6 possibilities.

What is the probability of rolling an*even*number with a single die? 3/6 or 1/2 because there are 3 even numbers [2,4,6], and 6 possibilities.

If you throw 5 heads in a row, what is the probability of throwing a head on the next flip? Well, the 6th flip is not at all dependent on the previous throws, so the probability is still 1/2.*This is a different question from asking what is the probability of throwing 6 heads in a row.*We will answer that in a minute.

**Values of probability**

Probability is expressed as a**fraction**: the denominator is the total number of ways things can occur and the numerator is the number of things that you are hoping will occur. Probability is always a number between 0 and 1 or between 0% and 100%. 0 means something cannot happen (impossible) and 1 (or 100%) means it is sure to happen.

**Single events**

Flipping coins and throwing a single die are examples of single events. Now for harder single events: How many cards are in a standard deck of playing cards? [52]. So, what is the probability of drawing:- One particular card (say, the 3 of spades)? [1/52]
- Any card in a particular suit (say, diamonds)? [13/52 or 1/4]
- Any card of a particular rank (say, a king)? [4/52 or 1/13]
- A red or a black card? [52/52 or 1, because all cards are either red or black]

**Two or more things happening**

Frequently we want to know the probability of 2 things happening, in other words, one thing happens AND then another thing happens. (AND means multiply). You*multiply*the probability of one thing happening by the probability of the other thing happening. What is the probability of:- Flipping 2 heads in a row? [1/2 x 1/2 = 1/4]
- Flipping 6 heads in a row? [1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/64] (pretty small!)
- Throwing 2 sixes on a die? [1/6 x 1/6 = 1/36]

(Rolling one die twice is the same as rolling 2 dice together once) - Throwing an 11 using 2 dice? [2/36 because there are 2 ways of throwing an 11: 5 + 6 and 6 + 5]
- Drawing 2 aces? It depends on if you put the 1st one back
before drawing the 2nd.

If you did put it back, then it is 4/52 x 4/52 = 16/2704 = 1/169.

If you didn't put it back, then it is 4/52 x 3/51 = 12/2652 = 1/221. The second probability is 3/51 because there are only 3 aces left and 51 cards left after you successfully take out the first ace. You have a better chance of getting 2 aces if you put the first one back before drawing again.

**Something does not happen:**

Well, if a sure thing has a probability of 1, then the probability that something*does not*happen is 1 minus the probability that it does happen. What is the probability that you will NOT throw a 1 using one die? [1 - 1/6 = 5/6]

## THE SUM OF TWO DICE |