1 9/23/2015 MATH 224 – Discrete Mathematics Basic finite probability is given by the formula, where |E| is the number of events and |S| is the total number of possible outcomes. This definition assumes that each possibility is equally likely. In other words each possibility has probability 1/|S|. Basics of Discrete Probability So for example, if we want the probability of four coin tosses where the first two times the coin comes up heads followed by two tails, this set E has only one element. The set of all possible 4-coin tosses is 2 4. So the probability of two heads followed by two tails is 1/16. On Page 394 of the textbook, there is a container with four blue balls and five red balls. What is the probability of choosing a blue ball when selecting one ball? The answer is 4/9.

2 9/23/2015 MATH 224 – Discrete Mathematics Now consider what happens when rolling a fair die. The probability of rolling any of the values between one and six is 1/6. When rolling two dice the probabilities are: P(2) = P(12) = 1/36 since there is only one way to get a 2 (two ones). P(3) = P(11) = 2/36 = 1/18 since there are two ways of getting a 3 (2 and 1 or 1 and a 2) P(4) = P(10) = 3/36 = 1/12 What are the three ways of getting four? P(5) = P(9) = 4/36 = 1/9 P(6) = P(8) = 5/36 P(7) = 6/36 = 1/6 Why is the denominator 36? The sum of theses probabilities is 2(1/36)+2(2/36)+2(3/36)+2(4/36)+2(5/36)+(6/36) = 36/36 =1. Do the set of all probabilities always sum to 1? Probability

3 9/23/2015 MATH 224 – Discrete Mathematics Often counting the number of different events can be difficult or tricky. Consider Example 6 from our text. First determine the number of different 5-card hands in poker in order to get the denominator |S|. Since there are 52 cards, |S| = C(52,5) = 2,598,960. What is the probability of a full house, i.e. 3 of a kind plus 2 of a kind? For three cards of a kind we have C(4,3) and C(4,2) for two of a kind once the kind of card is fixed. Where does the 4 come from in these combinations? Since there are 13 different kinds of cards we have P(13,2) for choosing the kinds. P(13,2) = = 146 permutations. Here we have permutations because which kind has three and which has two matters, e.g., two aces and three kings is different than two kings and three aces. Then the total number of ways of getting a full house is P(13,2)C(4,3)C(4,2) = What is the probability of getting a full house? (3744/2,598,960 ≈ ) If you want to determine the probability of, for example either a full house or at least two aces, you cannot just add the probabilities together since it is possible for both to be true at the same time. You have to count the number of events in E 1 U E 2, the union. Probability

4 9/23/2015 MATH 224 – Discrete Mathematics The Monty Hall example from the book is often confusing. Here is a common statement of this puzzle. There are three doors, two with nothing behind them and one with a prize. You choose one and then the game show host opens one of the doors you did not choose and nothing is behind it. Should you change doors? Initially you have a 1/3 chance of choosing the prize. After Monty opens a door, how do your odds change? If he always selects a door with no prize behind it, that is equivalent to saying divide the doors into two sets. One with two doors and one with just one door. Of the two door set, he shows one of the doors, always choosing one without a prize. The two door set has a 2/3 probability of having the prize and you know which one not to choose. You should change to the other unopened door. What if he chooses a door at random? Does this have an effect on the probabilities? Probability – Monty Hall Example

5 9/23/2015 MATH 224 – Discrete Mathematics Here are a few examples of the use of probability in computer science. In problems involving computer simulations of real events, probabilities are involved to specify events. For example, creating climate models. In analyzing the results of computer simulations, statistics and probability are used. In analyzing traffic and other parameters such packet sizes for computer networks, probability comes into play. Probability theory is essential in artificial intelligence. Often it is not possible to determine the best treatment, e.g., in diagnosing a medical problem. But probabilities can be used to help make a good choice of treatments. Probabilities are used to design algorithms. Some algorithms such as quick-sort sometimes use a probabilistic selection of an element. A data structure such as skip lists use probabilistic choices. Probability and statistics are used to analyze the average running time of algorithms. Applications of Probability to Computer Science

6 9/23/2015 MATH 224 – Discrete Mathematics First moment or Expected Value The first moment, often called expected value, is used to find the average running time for an algorithm. (This also referred to as the waited average.) The expected value is defined as, where X is the value at s. So for example, to find the average running time for an algorithm we would have where A is the average time, T(s) is the time for input s, and p(s) the probability of input s. As an example, let us consider insertion sort. We would need to know the probabilities that the array is sorted to begin with, that exactly x elements are in the wrong position, etc. Doing this precisely is difficult, but often since we only care about Big-O or Theta, it is often possible to make simplifying assumptions.

7 9/23/2015 MATH 224 – Discrete Mathematics Second Moment or Variance The second moment, often called the variance, indicates how much individual events vary from the average, and is given by The square root of the variance is called the standard deviation, which is a very important concept in statistics and for probabilities that fit the mathematical definition of a Normal distributions. For example, with this type of distribution just over 68% of the events will have values between ± 1 standard deviation from the mean. As an example, an instructor might decide that any grade within one standard deviation of the average is a C.