Chapter 15. Definitions: experiment: is the act of making an observation or taking a measurement.

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1 MATH 11008: Probability Chapter 15 Definitions: experiment: is the act of making an observation or taking a measurement. outcome: one of the possible things that can occur as a result of an experiment. sample space: set of all possible outcomes. event: any subset of the sample space. equally likely: two events are equally likely if they occur with equal relative frequency; equally often. Theoretical Probability of an event with equally likely outcomes: Suppose all outcomes in sample space S are equally likely. Let E be any event and n(e) = the number of outcomes in E and n(s)= the number of outcomes in S. Then the probability of event E, denoted P (E), is P (E) = n(e) n(s) Properties of Probability * For any event E, we have 0 P (E) 1. * For the empty set, P ( ) = 0. * For a sample space S, P (S) = 1. * E is an impossible event if P (E) = 0. * E is a certain event if P (E) = 1. * Let E represents the complement of E. Then P (E) = 1 P (E). * Let A B represents the union of A and B. Then P (A B) = P (A) + P (B) P (A B).

2 2 MATH 11008: PROBABILITY CHAPTER 15 Example 1: An urn contains 3 red balls, 2 blue balls, 5 yellow balls, and 6 black balls. If a ball is randomly chosen from the urn, find the probability that the ball is red? blue? black? Example 2: Consider the given spinner (a) Find the probability that it lands on an even number. (b) Find the probability that it does not land on a shaded number. (c) Find the probability that it lands on a shaded odd number. (d) Find the probability that it lands on a shaded number or an odd number.

3 MATH 11008: PROBABILITY CHAPTER 15 3 Example 3: Suppose you roll one red die and one blue die. What is the probability of getting a sum of 8 on a roll of these pair of dice? Example 4: Suppose that a card is chosen at random from a standard deck of playing cards. (a) What is the probability of drawing a spade? (b) What is the probability of drawing a queen? (c) What is the probability of drawing a queen of spades? (d) What is the probability of drawing a queen or spade? Example 5: Suppose two fair coins are tossed. Find the probability of each of the following. (a) Exactly one head (b) At least one head (c) At most one head

4 4 MATH 11008: PROBABILITY CHAPTER 15 Fundamental Counting Principle: Suppose a task consists of k separate parts. If the first part can be done in n 1 ways, the second part can be done in n 2 ways, and so on through the kth part, which can be done in n k ways, then the total number of ways to complete the task is given by the product: n 1 n 2 n 3 n k Example 6: Given the set of letters {a, b, c, d, e, f, g}. (a) How many four letters words can be made if the word must begin with a vowel and no repetitions are allowed? (b) How many four letters words can be made if the word must begin and end with a consonant and no repeats are allowed? (c) How many four letter words can be made if the word contains no vowels and no repeats are allowed? (d) How many four letter words can be made if the word must begin with a vowel and end with a consonant and repeats are allowed?

5 MATH 11008: PROBABILITY CHAPTER 15 5 Example 7: Consider the set of digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. (a) How many two digit numbers can be formed if repetitions are allowed? (b) How many two digit numbers can be formed if no repetitions are allowed? (c) How many three digit numbers can be formed if no repetitions are allowed? (d) How many three digit numbers can be formed if no repetitions are allowed, and the number must be odd?

6 6 MATH 11008: PROBABILITY CHAPTER 15 Tree Diagrams: Tree diagrams can be used to help determine the outcomes you are interested in. Example 8: Suppose a container has 5 marbles: 1 white, 2 green, and 2 yellow. An experiment consists of drawing one marble noting its color and then drawing a second marble and noting the color without replacement. Find the sample space for this experiment. Probability tree diagrams can be used to help to determine that probability of an outcome(s) of more complex experiments. Multiplicative Property of Probability Tree Diagrams: Suppose that an experiment consists of a sequence of simpler experiments that are represented by branches of a probability tree diagram. Then the probability of any of the simpler experiments is the product of all the probabilities on its branch.

7 MATH 11008: PROBABILITY CHAPTER 15 7 Example 9: An experiment consists of drawing a marble at random from Box #1, noting its color, and placing this marble into Box #2. Then we randomly choose a marble from Box #2, and note it s color. At the beginning of the experiment, Box #1 contains 2 red marbles, 4 green marbles, and 2 blue marbles and Box #2 contains 2 green marbles, and 3 blue marbles. (a) Draw a probability tree diagram for this experiment. (b) What is the probability of drawing two marbles the same color?

8 8 MATH 11008: PROBABILITY CHAPTER 15 Odds: Odds compares the number of favorable outcomes to the number of unfavorable outcomes. Suppose all outcomes in a sample space are equally likely where f of them are favorable to the event E and the remaining u outcomes are unfavorable to the event E. * Odds in favor of E: are f to u, denoted f : u. In other words, n(e) : n(e) * Odds against: are u to f, denoted u : f. In other words, n(e) : n(e) Odds ratio are normally simplified. 12 : 1 rather than 48 : 4. For example, it is preferable to express odds as Example 10: A local baseball team has won 13 games and lost 2 games. (a) What is the baseball team s odds in favor of winning the next game? (b) What is the baseball team s odds against winning the next game?

9 MATH 11008: PROBABILITY CHAPTER 15 9 Example 11: A card is drawn at random from a standard deck. Find (a) Odds in favor of drawing a face card. (b) Odds against drawing a diamond. (c) Odds in favor of drawing the ace of spades. (d) Odds against drawing a 2, 3 or 4. Example 12: Suppose that the odds in favor of an event E are 3 : 7. Find P (E). Example 13: If the P (E) = 5, find the odds against event E. 11

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