4.3 Rules of Probability


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1 4.3 Rules of Probability If a probability distribution is not uniform, to find the probability of a given event, add up the probabilities of all the individual outcomes that make up the event. Example: Suppose you are given the following probability distribution for a sample space S = {s 1, s 2, s 3, s 4, s 5, s 6 } Outcome s 1 s 2 s 3 s 4 s 5 s Probability Supppose E = {s 1, s 4, s 5 }, F = {s 2, s 3 }, and G = {s 2, s 5 }. Calculate the following. P (s 4 ) P (E) P (F G) P (E F ) P (E c ) P (E G) Example: Suppose an experiment has a sample space S = {s 1, s 2, s 3 } where P (s 1 ) + P (s 2 ) = 0.35 and P (s 2 ) + P (s 3 ) = Find the probability distribution for S. 1
2 What happens if you cannot list out all the outcomes and their probalities (or do not want to)? Or worse, what if we don t even know what the specific outcomes in the events are? We can use the following more general rules. Rules of Probability: 1. 0 P (E) 1 for any event E in a sample space S. In particular P ( ) = 0 and P (S) = Union rule for probability: If E and F are ANY two events, then P (E F ) = P (E) + P (F ) P (E F ) Note: If E and F are mutually exclusive, then E F =, which means P (E F ) = 0 and the formula reduces to just P (E F ) = P (E) + P (F ). 3. Complement Principle: P (E c ) = 1 P (E) or P (E) = 1 P (E c ) Example: Let E and F be two events of an experiment with sample space S. Suppose P (E) = 0.5, P (F ) = 0.4, and P (E F ) = 0.1. Compute the following. P (F c ) P (E c ) P (E F ) Example: If P (E c ) = 0.3 and P (F ) = 0.2 with E and F mutually exclusive, what is P (E F )? Example: In a survey of a group of people it was found that the probability someone did not like Dr. Pepper was 0.65, the probability someone liked Dr. Pepper was 0.45, and the probability that someone liked both Dr. Pepper and Coke was What is the probability that someone in this group likes Dr. Pepper or likes Coke? 2
3 Example: An experiment consists of selecting a card at random from a 52card deck. Find the probability that a red face card is drawn. Find the probability that a face card is not drawn. Find the probability that a diamond or a club is drawn. Find the probability that a spade or a queen is drawn. Find the probability that a 3 or a red card is drawn. Example: The table below gives the number of students of each classification who are majoring and not majoring in business in a class of 110 students. Freshmen Sophomores Juniors Seniors Total Business NonBusiness Total A student is randomly selected from this class. What is the probability that... The student is not a junior? The student is a freshman and a nonbusiness major? The student is a business major or a sophomore? The student is a nonbusiness major or is not a junior? 3
4 4.4 Random Variables and Expected Value A random variable is a rule that assigns a number to each outcome of an experiment. We usually denote a random variable by X. Example: A coin is tossed three times and the sequence of heads and tails is observed. List the outcomes of the experiment. Let the random variable X denote the number of tails that occur. What are the possible values of X? Find the probability distribution of X. What is P (1 X 2)? What is P (X > 0)? 4
5 The expected value of a random variable X, denoted E(X), is given by E(X) = x 1 p 1 + x 2 p x n p n where x 1, x 2,..., x n are the values that X may assume, and p 1, p 2,..., p n are the probabilities of each of these values. Example: Find the expected value of the random variable X given below. X Probability Consider the experiment of rolling 2 fair 5sided dice. Let X be the sum of the numbers rolled. Find the probability distribution of X. What is E(X)? Example: A survey was conducted of families to determine the distribution of families by size. The results are: Family Size Number of Families Let the random variable X be the number of people in a randomly chosen family. Find the probability distribution for X. What is the expected number of people in a randomly chosen family? 5
6 Expected values are often used in games to determine whether the game is fair. A game is considered fair when the expected NET winnings are 0. You are playing a game at a carnival. The game costs $1. You select a card from a standard deck. If the card is an ace, then you win $3. If the card is a face card, you win $2. Otherwise you win nothing. Find the expected net winnings. Example: A raffle is held people buy a ticket for $3 each. First prize is $1500. Second prize is $750. Then, four $100 consolation prizes will be given. What are the expected net winnings for one person who buys a $3 ticket. A game consists of rolling a fair 5sided die. The game costs $3 to play. If you roll an odd number, you win an amount of money equal to the number rolled. Otherwise you win nothing. What are your expected net winnings? 6
7 A game consists of rolling a pair of fair 6sided dice. The game costs $4 to play. If you roll a double, you win $a. Otherwise, you get nothing. What value of a would make this game fair? Example: A man purchased a $25,000 life insurance policy from his employer for $200/yr. (The cost of $200 is called the premium.) The probability that he lives another year is What is the life insurance company s expected net gain? If the probability that the man lives drops to 0.98, what is the minimum amount of money, $a, he can expect to pay for his policy? Note: The insurance company will charge at minimum an amount of money so that their expected net gain is 0. They would probably want to charge more than that to have a positive expected net gain. 7
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