## Notes on Probability

• 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]