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**A Survey of Probability Concepts**

Lesson 3

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**GOALS Define probability.**

Explain the terms experiment, event and outcome. Describe the classical, empirical, and subjective approaches to probability. Calculate probabilities using the rules of addition, complement rule, joint probability, rules of multiplication and conditional probability. Apply contingency table and tree diagram to organize and compute probabilities. Bayes’ theorem. Explain and apply multiplication, permutations, and combinations.

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Definitions A probability is a measure of the likelihood that an event in the future will happen. It it can only assume a value between 0 and 1. A value near zero means the event is not likely to happen. A value near one means it is likely.

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**Probability Examples P(Will not rain in Singapore this year) = 0**

P(Getting ‘1’ in a single toss of a die) = 1/6 P(Getting ‘tails’ in a single toss of a coin) = 0.5 P(Employee reporting to work today) > 0.8 P(Will rain in Singapore this year) = 1.0 Try giving some more examples.

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Definitions An experiment is the observation of some activity or the act of taking some measurement. An outcome is the particular result of an experiment. An event is the collection of one or more outcomes of an experiment.

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**Experiments, Events and Outcomes**

Tossing coins Rolling dices Experiments Outcomes 2 x ‘Head’ 1x‘Tail’ + 1x‘Head’ 1x‘Head’ + 1x‘Tail’ 2 x ‘Tail’ 2 x ‘6’ 1x‘5’ + 1x‘6’ … … 1x‘1’ + 1x‘2’ 2 x ‘1’ Events Observe 2 x ‘Head’ Observe 2 x (1x‘Head’+ 1x‘Tail’) Observe 3 outcomes whereby the sum of both die is more than 10

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**Assigning Probabilities**

There are three approaches of assigning probability: classical, empirical, subjective.

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**Classical Probability**

Consider an experiment of rolling two six-sided die. What is the probability of the event “the sum of both die is more than 10”? P(sum of both die > 10) = 3/36 The possible outcomes are: *Students to fill in the 36 possible outcomes as homework. There are 3 possible outcomes in the collection of 36 equally likely possible outcomes.

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**Mutually Exclusive, Independent, Collectively Exhaustive**

Events are mutually exclusive if the occurrence of any one event means that none of the others can occur at the same time. Events are independent if the occurrence of one event does not affect the occurrence of another. Events are collectively exhaustive if at least one of the events must occur when an experiment is conducted.

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**Empirical Probability**

The probability of an event happening is the fraction of the time similar events happened in the past. The key to establishing probabilities empirically is that more observations will provide a more accurate estimate of the probability. The empirical approach to probability is based on what is called the law of large numbers. Law of Large Numbers: Over a large number of trials the empirical probability of an event will approach its true probability.

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Law of Large Numbers Suppose we toss a fair die. The result of each roll is either ‘1’, ‘2’, ‘3’, ‘4’, ‘5’ or ‘6’. If we roll the die a great number of times, the probability of each outcome will approach 1/6. The following table reports the results of an experiment of rolling a fair die 1, 10, 50, 100, 500, 1,000, 5000 and 10,000 times and then computing the relative frequency of ‘1’. Note that this result is not repeatable. Why? *Try repeating the experiment for ‘2’, up to 100 times. Number of Trials Number of '1' Relative Frequency of '1' 1 10 2 50 8 100 17 500 82 1000 165 5000 850 10000 1655

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**Empirical Probability**

In 2006, there were floods experienced in parts of Singapore, after exceptional heavy rainfall. This was the second time in past 90 rainy days. On the basis of this information, what is the probability that a future rainy day would cause floods?

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**Subjective Probability**

Subjective Concept of Probability: The probability of a particular event happening that is assigned by an individual based on whatever information is available. If there is little or no past experience or information on which to base a probability, it may be arrived at subjectively. Examples of subjective probability are: 1. Estimating the likelihood the Singapore Soccer Team make it to World Cup. 2. Estimating the likelihood that your neighbour will be married before the age of 30. 3. Estimating the likelihood the Singapore budget surplus would exceed $5B next year.

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**Summary of Types of Probability**

Approaches to Probability Subjective Objective Classical Probability Empirical Probability Based on equally likely outcomes Based on relative frequencies Based on available information

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**Probability: Rules of Addition**

Special Rule of Addition If two events A and B are mutually exclusive, the probability of one or the other event’s occurring equals the sum of their probabilities. P(A or B) = P(A) + P(B) General Rule of Addition If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula: P(A or B) = P(A) + P(B) - P(A and B) Event A Event B Event A Event B P(A and B)

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**1 queen of clubs out of 52 cards**

Addition Rule What is the probability that a card chosen at random from a standard deck of cards will be either a queen or a club? P(A or B) = P(A) + P(B) - P(A and B) = 4/ /52 - 1/52 = 16/52, or .3077 Club Queen Queen of Club Card Probability Explanation Queen P(A) = 4/52 4 queens out of 52 cards Club P(B) = 13/52 13 clubs out of 52 cards Queen of Club P(A and B) = 1/52 1 queen of clubs out of 52 cards

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Complement Rule The complement rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1. P(A) + P(A’) = 1 or P(A) = 1 - P(A’). A A’

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**Joint Probability JOINT PROBABILITY:**

Probability that two or more events will happen concurrently. Club Queen Queen of Club

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**Probability: Rule of Multiplication**

Special rule of multiplication 1. Two events A and B are independent. 2. Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other. P(A and B) = P(A).P(B)

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**Conditional Probability**

Conditional probability is the probability of a particular event (A) occurring, given that another event (B) has occurred. When Event A occurs after Event B, and Event B has an effect on the likelihood that Event A occurs, then A and B are dependent. The probability of Event A given that Event B has occurred is written P(A|B).

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**General Multiplication Rule**

The general rule of multiplication is used to find the joint probability that two events will occur, when the events are not independent. It states that for two events, A and B, the joint probability that both events happening is found by multiplying the probability that Event A, P(A), will happen by the conditional probability of Event B occurring given that A, P(B|A), has occurred.

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**General Multiplication Rule**

There are 12 coloured balls in a bag. 9 of these balls are red and the others green. If we would to draw two balls consectively, without replacement. What is the likelihood both balls selected are red? The event that the first ball selected is red is R1. The probability is P(R1) = 9/12 The event that the second ball selected is also red is identified as R2. The conditional probability that the second ball selected is red, given that the first shirt selected is also red, is P(R2 | R1) = 8/11. To determine the probability of 2 red ball being selected we use formula: P(AnB) = P(A).P(B|A) P(R1 and R2) = P(R1).P(R2 |R1) = (9/12)(8/11) = 0.55

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Contingency Tables Contingency Table used to classify sample observations according to two or more identifiable characteristics. Number of Credit Cards (B1) 1 (B2) 2 (B3) 3 or more (B4) Total Men (A1) 4 3 12 Women (A2) 6 5 18 7 9 8 30 What is the probability of a randomly selected person is a women and does not have any credit cards? Event B1 happens if a randomly selected person does not have any credit cards. P(B1) = 6/30, or 0.2. Event A2 happens if a randomly selected person is a women. P(A2) = 18/30, or 0.6. Of the 18 women, 4 do not have credit cards. P(B1 | A2) is the conditional probability that women do not have any credit cards, P(B1 | A2) = 4/18 P(A2 and B1) = P(A2).P(B1 | A2) = 18/30 * 4/18 = 4/30

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Tree Diagrams A tree diagram is useful for illustrating conditional and joint probabilities. It is particularly useful for analyzing decisions and organizing calculations involving several stages. Each segment is one stage of the problem. The branches of a tree diagram are weighted probabilities.

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**Tree Diagram Men Women 1 credit card 2 credit card >3 credit card**

Conditional Probabilities 2/12 3/12 4/12 4/18 3/18 6/18 5/18 Gender Credit Card Joint 12/30 18/30 12/30 * 2/12 = 2/30 12/30 * 4/12 = 4/30 12/30 * 3/12 = 3/30 18/30 * 4/18 = 4/30 18/30 * 3/18 = 3/30 18/30 * 6/18 = 6/30 18/30 * 5/18 = 5/30 Must Be Equal to 1.00 Read textbook pages 158 and 159.

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Bayes’ Theorem Bayes' theorem relates the conditional and prior probabilities of random events. Prior Probability: The initial probability based on the present level of information. Posterior Probability: A revised probability based on additional information.

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Bayes’ Theorem WHO suspected that 5% of Country A’s population are infected with flu. A1: Event that person has flu P(A1) = 0.05 A2: Event that person does not have flu P(A2) = 1-P(A1) = 0.95 There is a diagnostic test available to detect the disease, but its accuracy is 90%. Event that person has flu and detected by test P(B|A1) = 0.9 Event that person does not has flu, but detected by test P(B|A2) = 0.15 When we randomly select a person and he is tested positive. What is the probability the person actually has flu? Find, P(person has flu | test is positive) Symbolically, P(A1|B) *Read textbook pages *Students to draw out the Tree Diagram for this example as homework.

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Bayes’ Theorem The above example illustrated the application of Bayes’ Theorem to only two mutually exclusive and collectively exhaustive events, A1 and A2. If there are n such events, A1, A2, …, An, Bayes’ Theorem becomes:

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Multiplication The multiplication formula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both. Multiplication Formula: Total number of arrangements = (m)(n) Example: A wife told her husband that she needed 10 sets of clothing, so that she would be able to ‘survive’ a two week cycle, without repeating the same set of clothes. Her husband disagreed and said that the same could be achieved with the following: 5 shirts and 2 pants/skirts 2 shirts and 5 pants/skirts - Do you agree?

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Permutation Permutation: Any arrangement of r objects selected from a single group of n possible objects. The order of arrangement is important in permutations. Permutation Formula: Where: n is the total number of objects. r is the number of objects selected.

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Permutation During a S-League match, Coach A had to select 3 players as ‘defender’, out of 11 players. How many ways can the 11 players be arranged in the 3 ‘defender’ positions?

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Combination Combinations: The number of ways to choose r objects from a group of n objects without regard to order. Combination Formula: Where: n is the total number of objects. r is the number of objects selected.

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Combination There are 18 players on the Foxtrot soccer team. Coach A must pick 11 players among the 18 players to comprise the starting team. How many different groups are possible?

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**End of Lesson 3 Refer to Textbook Chapter 5**

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