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Susan Kamp

Ch 14 From Randomness to Probability Many things in our world depend on randomness: - What is the chance to observe a Head while flipping a coin? - What is the chance to roll a 6 with a "fair" die? - What is the probability that the bus will be on time today? - What is the probability that you lose your investment? Even those events occur randomly, there is an underlying pattern in the occurrence of these events. This is the basis of Probability Theory. Definition: 1. A phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.  Individual outcomes are unpredictable  With a large number of observations, predictable patterns occur Examples for random phenomenon are:  Count red blood cells in a blood sample  Roll a die 1 of 30  Toss a coin  Toss a coin twice 2. In general, each occasion upon which we observe a random phenomenon is called a trial. 3. At each trial, we note the value of the random phenomenon, and call it an outcome. 4. When we combine outcomes, the resulting combination is an event. - An event is a subset of the sample space (or is the collection of one or more of the outcomes of an experiment) - Events are usually denoted by letters from the beginning of the alphabet, such as A and B, or by a letter or string of letters that describes the event. 5. The collection of all possible outcomes is called the sample space. Con’t Example: Give the sample spaces of above random phenomenon: 2 of 30 Examples for events: Suppose you roll an unbiased die with 6 faces and observe the number on the upper face. Let A be the event of rolling 1 or 2 Let B be the event of rolling 3 or 4 Let C be the event of rolling an odd number Example: the 2 coins toss. Let A be the event that occurs if at least a head is tossed. Example: If we toss 2 coins, what is the sample space? Draw a tree diagram to illustrate it. 3 of 30 This is a tree diagram (each outcome is represented by a branch of the tree).  An ideal way of visualizing sample spaces with a small number of outcomes  As the number of trials or the number of possible outcomes on each trial increase, the tree diagram becomes impractical Example: In a pop quiz with 3 multiple-choice questions, each question has 5 options, and the student’s answer is either correct (C) or incorrect (I). Determine the total number of possible outcomes. What is the sample space for the correctness of a student’s answers on this pop quiz. Draw a tree diagram to illustrate it. Let A be the event a student answers all 3 questions correctly Let B be the event a student passes (at least 2 correct) 4 of 30 6. The probability of any outcome (or event) of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions. (Probability is a measure for the likelihood or chance of a future event) - This definition is based on the Law of Large Numbers (LLN): thee long-run relative frequency of repeated independent events gets closer and closer to a single value. - The LLN says nothing about short-run behavior. - Relative frequencies even out only in the long run, and this long run is really long (infinitely long, in fact). Equally likely - It’s equally likely to get any one of six outcomes from the roll of a fair die. - It’s equally likely to get heads or tails from the toss of a fair coin. - However, keep in mind that events are not always equally likely. o A skilled basketball player has a better than 50-50 chance of making a free throw. 5 of 30 Probability of an Event with equally likely outcomes The probability of an event (with equally likely outcomes) is the number of outcomes in the event divided by the total number of possible outcomes. Number of times A occurs P(A)  #of possible outcomes - P(A) = sum of the probabilities of the individual outcomes contained in A. Example: Find the probability of A. 1)Let A be the event of obtaining a ‘2’ in one roll of an unbiased die. 2)Let A be the event of obtaining a ‘H’ in one toss of an unbiased coin. 3)Let A be the event of obtaining a ‘HH’ in two tosses of an unbiased coin. 4)Let A be the event of obtaining an even number in one roll of an unbiased die. 6 of 30 5)Let A be the event of obtaining at least one tail in two tosses of an unbiased coin. A Venn diagram is a picture that depicts all the possible outcomes for an experiment. The outer box represents the sample space, which contains all of the outcomes, and the appropriate events are circled and labeled. Example: Draw a Venn Diagram for the dice example: Recall: - A = {1, 2} - B = {3, 4} - C = {1, 3, 5} The probabilities must follow the following rules: 1) 2 requirements for a probability: - The probability of each simple event (individual outcome) is between 0 and 1. ie. For any event A, 0  P(A)  1 - P(A) = 0, if the event A never occurs. 7 of 30 - P(A) = 1, if the event always occurs. 2) Probability Assignment Rule: the probability of the set of all possible outcomes of a trial must be 1. ie. then P(S) = 1, where S is the whole sample space or the set of all possible outcomes - Roll a 6-sided die: P({1,2,3,4,5,6}) = - P(roll with a regular die a number < 7) = - For the toss of 1 coin: P(S) = P({H, T}) = - For the toss of 2 coins: P(S) = P({HH,HT,TH,TT})= Example: Blood Types All human blood can be typed as one of O, A, B, or AB, but the distribution varies a bit with race. Here is the model for a randomly chosen black American. Blood Type O A B AB Pbty 0.49 0.27 0.2 ? a) What is the probability of type AB blood? Why? 8 of 30 b)Maria has type B blood. She can safely receive blood transfusions from people with blood types B and O. What is the chance that a randomly selected black American can donate blood to Maria? 3. Complement Rule: a. Complement of an event A (denoted by A ) c a. Consists of all outcomes in the sample space that are not in A. b. The probabilities of A and A add to 1 (ie. P(A ) + P(A) = 1) c. P(A ) = 1 – P(A) Remark: P(A  A ) = 0c P(A  A ) = 1 Example 2 cont: Using the pop quiz example, draw a Venn diagram to illustrate B and B . Recall: B is the event that a student passes (at least 2 correct); 9 of 30 B = {CCI, CIC, ICC, CCC} B = {CII, IIC, ICI, III} NOTE: B is the event that a student fails (less than 2 correct); Example: Use the complement rule to find the probability of having at least one tail in two tosses of an unbiased coin? Recall: P(A) = P({TH, HT, TT}) = 3/4 OR P(A) = 1 – P(A ) = 1 – P(no tails in the two tosses) = 1 – P({HH}) = 1 – ¼ = ¾ Definitions: - If A and B are disjoint events (or mutually exclusive events), then they have no outcomes in common (ie. if when one event occurs, the other cannot occur, and vice versa.) 10 of 30 Example Cont: Quiz Recall: A: student answers all 3 questions correctly B: student passes (at least 2 correct) Now, let • Event C: Student answers exactly 1 question correctly • Event D: Student answer exactly 2 questions correctly Which of the events (A, B, C, D) are disjoint? In another words: A = {CCC} B = {CCI, CIC, ICC, CCC} C = {CII, ICI, IIC} D = {CCI, CIC, ICC} Notations: A  B = A and B A  B = A or B Let A and B be two events: - The intersection of events A and B, denoted by A  B, consists of outcomes that are in both A and B. 11 of 30 - The union of events A and B, denoted by A  B, is the event that either A or B or both occur. General Addition Rule: Probability of the Union of 2 Events P(A  B) = P(A) + P(B) – P(AB) Addition Rule: If the events are disjoint: P(A  B) = P(A) + P(B) Remark 1: The subtraction of P(AB) is necessary because this area is counted twice by the addition of P(A) and P(B), once in P(A) and once in P(B). Remark 2: The probability of the intersection of disjoint events is P(A∩B) = 0 Example: A fair dice Let A be the event to roll an odd number. Let B be the event to roll a number greater than 2. What is P(A), P(B), P(A ), P(B ), P(AB), P(AB)? P(A) = P({1,3,5}) = c c P(A ) = 1 - P(A) = OR: P(A ) = P({2,4,6}) = 12 of 30 P(B) = P({3,4,5,6}) = c c P(B ) = 1 – P(B) = OR: P(B ) = P({1,2}) = P(AB) = P({1,3,5}  {3,4,5,6}) = P(AB) = P({1,3,5}  {3,4,5,6}) = OR P(AB) = P(A) + P(B) – P(AB) Example: Color Suppose that 55% like green, 25% like red, and 45% like yellow. Also, suppose 15% like both green and red, 5% like all 3 colors, 25% like both green and yellow, and 5% only like red. How many like green or red or both? How to find P(A and B)? Sometimes P(AB) is not given in a question, and the rule for calculating P(AB) depends on the idea of inindependent and dependent events. 13 of 30 Multiplication Rule: Probability of the Intersection of Independent Events P(A and B) = P(A) x P(B) Independence of Two Events Two events are considered independent, when the occurrence of one of the events has no impact on the probability for the second event to occur. - NOTE: it doesn’t mean they are disjoint!!  Disjoint events cannot be independent! Well, why not?  Since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn’t.  Thus, the probability of the second occurring changed based on our knowledge that the first occurred.  It follows, then, that the two events are not independent.  A common error is to treat disjoint events as if they were independent, and apply the Multiplication Rule for independent events—don’t make that mistake. 14 of 30 Example: Toss a fair coin twice. What is the probability to toss two heads? Recall: P({HH}) = ¼ OR Define: A: head on second toss B: head on first toss This is independent event. Example: What is the probability to get the outcome (HTTH) with a biased coin which has a 0.1 chance to toss a head. P({HTTH}) = P(H)P(T)P(T)P(H) Example: Two and Three Dice Rolling Game. a) When you roll two dice, what is the probability to roll two 6’s? There are 36 possible outcomes: 1 1 1 2 1 3 1 4 1 5 1 6 2 1 2 2 2 3 2
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