2GOALS 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.
3DefinitionsA 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.
4Probability Examples P(Will not rain in Singapore this year) = 0 P(Getting ‘1’ in a single toss of a die) = 1/6P(Getting ‘tails’ in a single toss of a coin) = 0.5P(Employee reporting to work today) > 0.8P(Will rain in Singapore this year) = 1.0Try giving some more examples.
5DefinitionsAn 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.
6Experiments, Events and Outcomes Tossing coinsRolling dicesExperimentsOutcomes2 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’EventsObserve 2 x ‘Head’Observe 2 x (1x‘Head’+ 1x‘Tail’)Observe 3 outcomes whereby the sum of both die is more than 10
7Assigning Probabilities There are three approaches of assigning probability:classical,empirical,subjective.
8Classical 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) = 3/36The 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.
9Mutually 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.
10Empirical 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.
11Law of Large NumbersSuppose 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 TrialsNumber of '1'Relative Frequency of '1'1102508100175008210001655000850100001655
12Empirical 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?
13Subjective 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.
14Summary of Types of Probability Approaches to ProbabilitySubjectiveObjectiveClassical ProbabilityEmpirical ProbabilityBased on equally likely outcomesBased on relative frequenciesBased on available information
15Probability: Rules of Addition Special Rule of AdditionIf 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 AdditionIf 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 AEvent BEvent AEvent BP(A and B)
161 queen of clubs out of 52 cards Addition RuleWhat 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 .3077ClubQueenQueen of ClubCardProbabilityExplanationQueenP(A) = 4/524 queens out of 52 cardsClubP(B) = 13/5213 clubs out of 52 cardsQueen of ClubP(A and B) = 1/521 queen of clubs out of 52 cards
17Complement RuleThe 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’) = 1or P(A) = 1 - P(A’).AA’
18Joint Probability JOINT PROBABILITY: Probability that two or more events will happen concurrently.ClubQueenQueen of Club
19Probability: Rule of Multiplication Special rule of multiplication1. 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)
20Conditional 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).
21General 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.
22General 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) = 9/12The 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
23Contingency TablesContingency 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)TotalMen (A1)4312Women (A2)651879830What 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) = 4/18P(A2 and B1) = P(A2).P(B1 | A2) = 18/30 * 4/18 = 4/30
24Tree DiagramsA 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.
26Bayes’ TheoremBayes' 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.
27Bayes’ TheoremWHO suspected that 5% of Country A’s population are infected with flu.A1: Event that person has fluP(A1) = 0.05A2: Event that person does not have fluP(A2) = 1-P(A1) = 0.95There is a diagnostic test available to detect the disease, but its accuracy is 90%.Event that person has flu and detected by testP(B|A1) = 0.9Event that person does not has flu, but detected by testP(B|A2) = 0.15When 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.
28Bayes’ TheoremThe 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:
29MultiplicationThe 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/skirts2 shirts and 5 pants/skirts- Do you agree?
30PermutationPermutation: 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.
31PermutationDuring 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?
32CombinationCombinations: 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.
33CombinationThere 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?