Chap 4 Probability p227 The probability of any outcome in a random phenomenon is the proportion of times the outcome would occur in a long series of

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1 Chap 4 Probability p227 The probability of any outcome in a random phenomenon is the proportion of times the outcome would occur in a long series of repetitions. (p229) That is, probability is a long-term relative frequency. Example: Tossing a coin: P(H) =? Sample space p233 The sample space of a random phenomenon is the set of all possible outcomes. Example 4.4 p233 Toss a coin. Sample space: S = {H, T} Example: Rolling a die. S = {1, 2, 3, 4, 5,6} 1

2 Event p235 An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space. Example: The sample space (S) for two tosses of a coin to is {HH, HT, TH, TT}. Then exactly one head is an event, call it A, then A = {HT, TH}. Notation: The probability of an event A is denoted by P(A). 2

3 Equally likely outcomes p240 If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k. The probability of any event A is PA ( ) count of outcomes in count of outcomes in A S count of outcomes in A k Ex: A pair of fair dice is rolled. What is the probability that the 2 nd die lands on a higher value than does the 1 st? 3

4 Probability rules p The probability P(A) of any event A satisfies 0 PA ( ) If S is the sample space in a probability model, then P(S) = The complement of any event A is the event that A does not occur, written as A c. The complement rule states that P( Ac) 1 P( A). 4. Two events A and B are disjoint if they have no outcomes in common and so can never occur simultaneously. If A and B are disjoint, P( Aor B) P( A) P( B). This is the addition rule for disjoint events. - This can be extended for more than two events - Venn diagrams p236 4

5 Question Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement about an event. (The probability is usually a much more exact measure of likelihood than is the verbal statement.) 0; 0.01; 0.3; 0.6; 0.99; 1 (a) This event is impossible. It can never occur. (b) This event is certain. It will occur on every trial of the random phenomenon. (c) This event is very unlikely, but it will occur once in a while in a long sequence of trials. (d) This event will occur more often than not. 5

6 Question If you draw an M&M candy at random from a bag of the candies, the candy you draw will have one of six colors. The probability of drawing each color depends on the proportion of each color among all candies made. The table below gives the probability of each color for a randomly chosen plain M&M: Color Brown Red Yellow Green Orange Blue Probability ? a) What must be the probability of drawing a blue candy? b) What is the probability that a plain M&M is any of red, yellow, or orange? c) What is the probability that a plain M&M is not red? 6

7 Question Choose an American farm at random and measure its size in acres. Here are the probabilities that the farm chosen falls in several acreage categories: Let A be the event that the farm is less than 50 acres in size, and let B be the event that it is 500 acres or more. (a) Find P(A) and P(B). (b) Describe A c in words and find P(A c ) by the complement rule. (c) Describe {A or B} in words and find its probability by the addition rule. 7

8 Independent events p242 Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. Multiplication rule for independent events p242 If A and B are independent events, P( Aand B) P( A) P( B). 8

9 Ex: The gene for albinism in humans is recessive. That is, carriers of this gene have probability 1/2 of passing it to a child, and the child is albino only if both parents pass the albinism gene. Parents pass their genes independently of each other. If both parents carry the albinism gene, what is the probability that their first child is albino? If they have two children (who inherit independently of each other), what is the probability that a) both are albino? b) neither is albino? c) exactly one of the two children is albino? d) If they have three children (who inherit independently of each other), what is the probability that at least one of them is albino? 9

Probability and Sample space

Probability and Sample space We call a phenomenon random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions. The probability of any outcome