CHAPTERS 14 & 15 PROBABILITY STAT 203

Transcription

1 CHAPTERS 14 & 15 PROBABILITY STAT 203

2 Where this fits in 2 Up to now, we ve mostly discussed how to handle data (descriptive statistics) and how to collect data. Regression has been the only form of statistical inference that we ve seen thus far. Underlying almost all forms of statistical inference is probability. Most statistical inferences are based on probability models.

3 Randomness 3 Probability is the study of randomness. A random phenomenon is an experiment whose outcome is not known before hand. E.g. flipping a coin. However, for an experiment to be random the possible outcomes should be known beforehand. Example: When a tornado hits a town, the aftermath of the tornado is not known before hand, but we also have no idea what it can do no two tornadoes are the same. This is simply chaos. On the other hand, the number of deaths caused by a tornado is a random phenomenon. We don t know it before hand, but the possible outcomes are 0,1,2,3,4,

4 Concepts and Definitions 1 4 Every time we run a random phenomenon, such as flip a coin, we obtain a single outcome. An outcome is not random. A single run is referred to as a trial. An event is a collection of outcomes. A probability is the chance that a given event will occur. It is expressed as a relative frequency, so it is a number between 0 and 1. An event has probability p means that if the random phenomenon is repeated over and over, this event will occur 100(p)% of the time.

5 Card Playing 5 Consider a deck of cards containing 52 cards. If we draw one card from the deck, then the possible outcomes is each of the 52 cards. Ace of hearts, ace of spades, etc. Drawing the King of Diamonds can be considered both an outcome and an event. Drawing a red card is an event, but not an outcome.

6 Concepts and Definitions 2 6 The Sample Space (S) is the set of all possible outcomes from a random phenomenon. For a coin toss: S = {Heads, Tails} For rolling a die: S = {1, 2, 3, 4, 5, 6} We denote Events by upper case letters usually at the start of the alphabet. Let A = get heads Let B = draw a face card. The probability of an event A (say getting heads) occurring is denoted by P(A). It is acceptable to write P(get heads) for sake of clarity.

7 Properties of P(A) 7 For any event A from a sample space S 1. 0 P(A) 1 An event can occur between 0% and 100% of trials. The larger P(A), the more probable the event is of occurring. 2. P(S) = 1 An outcome in the sample space will occur. 3. The sum of the probabilities of all outcomes (nonoverlapping by definition) in the sample space sums up to one.

8 Establishing probabilities 8 (This is conceptual and serves for a greater understanding, you will not be asked to solve problems in this fashion) How do we determine the probability of an outcome or event in real life? In simpler scenarios, we can use physical considerations. E.g. flipping a coin has two outcomes and they are equally likely. In more complex cases we use past trials to estimate the probability.

9 The Law of Large Numbers 9 The word large here refers to the number of trials. The Law of Large Numbers states that as the number of trials increases, the relative frequency gets closer to the true probability. Example: If we flip a coin 10 times, it isn t unlikely that we get 4 or less heads, but if we flip a coin a million times, it is highly unlikely that we flip it 400,000 times or less. Thus it is valid to use relative frequency over a large number of trials as an estimate for probability

10 First steps towards math 10 Equally Likely Outcomes: Suppose the sample space is composed of equally likely outcomes. Then the probability of an event A occurring is E.g. Rolling a die: What is the probability of getting an even number?

11 Equally Likely Outcomes 11 Equally likely outcomes does not imply equally likely events. Getting an odd number or an even number at roulettes is equally likely, but winning and losing the lottery is not. Example in a hat are 5 red balls, 3 green balls and 2 blue balls. What is the probability of selecting a red ball? What is the probability of not selecting a blue ball?

12 Event operators 1 12 It is simplest to work with outcomes, but most often we work with events or even combination of events. E.g. get a green ball first and blue ball second. Due to this, it is useful to develop rules for events. These follow the notions of set theory. We will use the drawing a card from a deck example to illustrate this: A = Draw a red card B = Draw a jack C = Draw a face card D = Draw a club

13 Event Operators 2 13 Union: The union of two events A, B is denoted by and means A or B or both occur. E.g. in our example means the card drawn is either a jack, queen, king or another card of clubs. Intersect: The intersect of two events A, B is denoted by and means both A and B occur. E.g. In our example, means draw the jack of clubs, the queen of clubs or the king of clubs.

14 Event Operators 3 14 Compliment: The compliment of an Event A is the Event that A does not occur and is written A c E.g. the event that someone shows symptoms is A c The event that someone does not have a fever is B c Disjoint (or Mutually Exclusive): two events are disjoint if only one can occur at a time. A and B are disjoint, as are A and D, but B and D are not. Independence: Two events are independent if the occurrence/non-occurrence of one does not affect the probability of the other. (Consider card example)

15 15 Venn Diagram 1

16 16 Venn Diagram 2

17 Simple Probability Rules 1 17 Complement Rule: Suppose the probability of fever P(B)=0.23, then P(B c )= 1 P(B) = = 0.77 Note that this can save a lot of time, so be aware of when you can use it! Addition Rule: If two events A, B are disjoint, then the probability of A or B is the sum of the probabilities.

18 Simple Probability Rules 2 18 Multiplication Rule: If two events are independent, then the probability of A and B is the product of the event probabilities E.g. Suppose we flip a coin two times. The outcome in the first trial does not tell us anything about the outcome in the next trial, so the first flip and second flip are independent. Therefore the probability of getting two consecutive heads is P(H1 and H2) = P(H1) x P(H2) = 0.5 x 0.5 = 0.25

19 Cards 19 Consider the following events: A={Draw a red card} B= {Draw a black card} C= {Draw a Heart} D= {Draw a King} E= {Draw the Ace of spades} F={Draw higher than 10} For each of the following pairs of events, determine if they are disjoint and whether they are Independent. (A,B); (A,C); (B,D); (D,F); (E,C); (B,F); (E,D) What is P(A or B)?P(A and B)? P(D or E)? P(D or F)? P(D and F)?

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21 Early Risers 21 A study investigated the natural sleeping tendencies among couples. It found that in 35% of couples both partners were early risers, while in 20% of couples only the wife was an early riser and 45% of husbands are early risers. a) What is the probability that a wife is an early riser? b) What is the probability that a couple is in sleeping agreement? c) What is the probability that a couple has at least one early riser?

22 Back to equally likely events 22 Consider our hat with 5 red balls, 3 green balls and 2 blue balls. If we select two balls is the color of the second one independent of the color of the second one? Answer: It depends on whether we select with or without replacement. If we select a first ball and then place it back in the hat, then the probabilities for each color do not change in the second trial, but if we don t replace the ball, then

23 23 Chapter 15 Probability Rules

24 Conditional Probabilities 24 We use conditional probabilities when two events are not independent. We condition the probability on the occurrence/non-occurrence of an other event. Let A and B be two events such that P(A) 0. Then we denote the conditional probability that B will occur given that A has occurred by P(B A) and define

25 The Taste Test 25 Let A = {Bottle is Glass}, B={Coke is Preferred}, then: The probability of preferring coke if the bottle is made of glass is P(B A)= P(B and A) = 21/113 = 21 P(A) 32/ The probability that the bottle is made of plastic given that Pepsi is preferred is

26 Check for independence 26 It is important to distinguish the difference between the independence of variables and that of events. To verify if two events are independent, we should verify if P(B)=P(B A) What do we conclude about B and A in the Income and Tasting example?

27 The General Addition Rule 27 We ve seen the addition rule for events which are disjoint. If two events are not disjoint, then Of course, if two events are disjoint, then P(A and B)=0, which leads to the simple version seen earlier.

28 The General Multiplication Rule 28 We ve seen the multiplication rule for events which are independent. The multiplication rule for events which are not independent is Of course, if A and B are independent, then P(A B)=P(A) and P(B A)=P(B), so

29 Probability Trees 29 In the previous example, we were given information about intersects and asked bout conditional probabilities. In some occasions, we are given the conditional probabilities and asked to find the probabilities of the intersect or something more complex. In such cases, a probability tree may be useful.

30 Health Insurance 30 A health economist has established that: 5% of all adults have utilized emergency medical care in the last 12 months. Of the 5%, 80% had health insurance. It was also found that 55% of people who did not use emergency medical care had health insurance. If you randomly select someone, what is the probability that they have health insurance and did not use emergency medical care? What s the probability of having health insurance or not utilizing emergency medical care? What is the probability that a person has health insurance?

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33 Bayes Theorem 33 Bayes theorem when we have a conditional probability, say P(A B), but want the reverse conditional probability P(B A). The equation is P(B A) = P(A B)P(B) = P(A and B) P(A B)P(B)+P(A B c )P(B c ) P(A)

34 Example 34 The classic example is using diagnostic testing. Suppose there s a disease which 5 in 10,000 have. Suppose also that the test for the disease is positive for 99% of people who have the disease and negative for 99% of people who are disease free. Given that a person has tested positive (A), what is the probability that they have the disease (B).

35 Decision Making (some tips) 35 Want to solve a conditional probability based on other conditional probabilities Use Bayes Theorem Want to solve the probability of an event or intersect based on conditional probabilities Use Probability Trees If there is only one conditional probability Look to use the Multiplication Rule When given the intercept and probability of events or unions We ll often want to use Venn Diagrams