4.1 Sample Spaces and Events

Transcription

1 4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an outcome of the experiment. The sample space of an experiment is the set of all possible outcomes. Example: Determine the sample space, S, for the following experiments. Flipping a coin and observing whether it lands heads or tails. Rolling a fair 6-sided die and observing the number that is rolled. Rolling two fair 6-sided dice and observing the sum of the numbers rolled. An event is a subset of the sample space of an experiment. Example: An experiment consists of rolling two fair dice and observing the numbers rolled. The sample space S for this experiment is: S = (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) A shorthand way of working with the outcomes is: The first coordinate of these ordered pairs represents the first die and the second coordinate represents the second die. (3, 5) and (5, 3) are different outcomes. You can think of one die as red and the other as green. Determine the event E that the sum of the two dice is 6. Determine the event F that a double is rolled. Determine the event G that the sum of the dice is less than 5. Determine the event H that a 6 is rolled. Determine the event K that a 7 is rolled. The impossible event is denoted by or {} 1

2 There are three operations we can do with events: complements, unions, and intersections. If E and F are two events of an experiment, then: E c is the event that E does NOT occur. E F is the event that E OR F (or both) occur. E F is the event that BOTH E AND F occur. From the dice example above we saw that: E = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)} F = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} G = {(1, 1), (1, 2), (2, 1), (3, 1), (2, 2), (1, 3)} H = {(1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)} Using our dice example from above: What is the event H c? In other words, what is the event that a 6 is not rolled? What is the event F G? In other words, what is the event that a double is rolled OR the sum of the dice is less than 5? What is the event F G? In other words, what is the event that a double is rolled AND the sum of the two dice is less than 5? If two events A and B CANNOT happen at the same time, then A B =, and these events are said to be mutually exclusive. Are E and G mutually exclusive? Are E and F mutually exclusive? 2

3 Often, tree diagrams can be used to help find the sample space. Example: Suppose I flip a coin twice and record the side that lands up on each toss. Determine the sample space for this experiment. Determine the event E that a head is not tossed. Determine the event F that a head is tossed and a tail is tossed. Are E and F mutually exclusive? A note on decks of cards: A deck of cards consists of 52 cards. There are 13 cards for each of the four suits: clubs, spades, diamonds, and hearts. The 13 cards are numbered 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace. Clubs and spades are black. Diamonds and hearts are red. A face card is a Jack, Queen, or King. (An Ace is NOT considered a face card.) Example: A fair 5-sided die is rolled, observing the number rolled, and then a card is selected from a standard deck, observing the color of the card. Determine the sample space for this experiment. Determine the event E that a 3 is rolled. Determine the event F that an even number is rolled or a black card is selected. Determine the event G that an odd number is rolled and a red card is selected. Are F and G mutually exclusive? 3

4 Example: A letter is selected at random from the word MATH, observing if it is a vowel or not, and then a card is randomly selected from a standard deck, observing the suit of the card. What is the sample space for this experiment? Determine the event E that a heart is selected. Determine the event F that a vowel is drawn or a black card is selected. Are E and F mutually exclusive? Example: An experiment consists of selecting a letter at random from the word HELLO, observing the letter chosen, rolling a fair 6-sided die and observing whether the number rolled is even or odd, and then randomly selecting a marble from a jar containing 3 yellow, 2 green, and 4 white marbles and observing the color of the marble. How many outcomes would be in the sample space for this experiment? List one possible outcome. 4

5 4.2 Basics of Probability Definition: A sample space S in which all outcomes are equally likely is called a uniform sample space. If S is a finite uniform sample space and E is any event, then the probability of E, P (E), is given by: P (E) = Number of ways for E to occur Total number of possible outcomes in S = n(e) n(s) Note: Probabilities will ALWAYS be between 0 and 1, inclusive. The larger the probability, the more likely it is to occur. Example: Suppose a fair die is rolled and the number that lands up is recorded. The sample space for this experiment is S = {1, 2, 3, 4, 5, 6}. Is this a uniform sample space? What is the probability that an even number is rolled? What is the probability that a number less than 3 is rolled? Example: A card is drawn from a standard deck of cards. What is the probability that: a Jack is drawn? A club? A face card? Example: Consider the experiment of rolling two fair dice and observing the numbers that land up. We already found the sample space. What is the probability that the sum of the dice is more than 10? What is the probability that a 5 is rolled? What is the probability that a double is not rolled? What is the probability that exactly one 4 is rolled or the sum of the dice is 5? 5

6 Example: Consider the composition of a three-child family in which the children were born at different times. Assume that a girl is as likely as a boy at each birth. What is the sample space for this experiment? What is the probability that there is exactly 1 boy in the family? What is the probability that there are at least two boys in the family? Sometimes experiments are run to help estimate the probability of certain events. Probabilities that are based on collected data are called empirical probabilities. If an experiment is performed n times and an event E occurs m times, then the relative frequency of the event E is m n. Example: In a survey conducted to see how long Americans keep their cars, a group of 2000 car owners were asked how long they plan to keep their present cars. The results are: Years Car is Kept, x Respondents 0 x < x < x < x < x < x What is the empirical probability that a randomly selected car owner in America plans to keep his/her car less than 4 years? At least 6 years? The probability distribution for an experiment is a table which gives the probabilities associated with events in the experiment. A probability distribution must satisfy the following properties: The events listed must be mutually exclusive. (If each outcome is listed separately, this will be satisfied.) The sum of the probabilities must be 1. 6

7 Example: Suppose I have a jar filled with 4 red marbles, 2 blue marbles, and 7 white marbles. An experiment consists of selecting one marble from the jar and observing its color. Find the probability distribution for the color of the marble. What is the probability that the marble is not red? Example: Find the probability distribution for the sum of the dice when two fair 6-sided dice are rolled. Example: An unfair 6-sided die is rolled over and over and the number rolled each time is recorded. The results are given below. Number Rolled Frequency Find the empirical probability distribution for this data. What is the empirical probability that an odd number is rolled? 7

8 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. 8

9 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 Pepsi was 0.65, the probability someone liked Coke was 0.45, and the probability that someone liked both Pepsi and Coke was What is the probability that someone in this group likes Pepsi or likes Coke? 9

10 Example: An experiment consists of selecting a card at random from a 52-card 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 Non-Business 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 non-business major? The student is a business major or a sophomore? The student is a non-business major or is not a junior? 10

11 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)? 11

12 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 5-sided dice. We know that the sample space of this experiment is 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? 12

13 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 5-sided 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? 13

14 A game consists of rolling a pair of fair 6-sided 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. 14