Chapter 6: Probability The Study of Randomness 6.1 The Idea of Probability 6.2 Probability Models 6.3 General Probability Rules 1
Simple Question: If tossing a coin, what is the probability of the coin turning up heads? Most of you probably answered 50%, but how do you know this to be so? 2
Probability Probability is the branch of mathematics that describes the pattern of chance outcomes. The heart of statistics, and thus the heart of this course, is in statistical inference. Probability calculations are the basis for inference. Mathematical probability is an idealization based on imagining what would happen in an indefinitely long series of trials. 3
Random Behavior Many observable phenomena are random: the relative frequencies of outcomes seem to settle down over the long haul. The big idea of probability: chance ( random ) behavior is unpredictable in the short run, but has regular and predictable patterns in the long run. See Example 6.1, p. 331 4
Figure 6.1, p. 331 5
Simulation Problem Problem 6.3, p. 334 6
Probability Definitions Sample Space (S) set of all possible outcomes. See Example 6.3, p. 336 (rolling two dice) Event a particular outcome or set of outcomes; a subset of the sample space. Probability the number of times an event can occur within the sample space divided by the sample space. What is the probability of getting a sum 5 when rolling two dice? 7
Figure 6.2, p. 336 8
Multiplication Principle Multiplication (counting) principle If you can do one task in a ways and a second task in b ways, then both tasks can be done in a x b ways. How many outcomes are possible if we flip 4 coins? Apply the multiplication rule to the 2-dice problem. Example 6.5, p. 337 Tree diagram 9
Figure 6.3, p. 338 10
HW Exercises 6.11, p. 340 6.14 a, b p. 341 6.18, p. 342 Reading, pp. 335-352 11
Probability Rules (Box on p. 343) Rule 1. If P(A) is the probability of an event A, then: 0 P( A) 1 Rule 2. If S is the sample space in a probability model, then: P( S) = 1 Rule 3. The complement (A c ) of an event A is the event that A does not occur. The complement rule states: P( A c ) = 1 P( A) 12
Probability Rules, cont. (Box on p. 343 and p. 351) Rule 4. Two events are disjoint (mutually exclusive) if they have no outcomes in common and can never occur simultaneously. If A and B are disjoint, then: P(A or B) = P(A) + P(B) Rule 5. Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent, then: P ( A and B) = P( A) P( B) 13
Independence Example What is the probability of drawing two hearts on successive draws from a standard deck of playing cards? With replacement Without replacement 14
Set Notation A or B = { A B}...read,A union B. Aintersect B = { A B} For mutually exclusiveevents: { A B} = (empty set) 15
In-Class Problems 6.19, p. 348 Look at example 6.10, p. 345 6.26, p. 349 More Problems: 6.27, 6.28, p. 354 6.31, 6.32, 6.33, p. 355 16
HW Problems: 6.36, 6.37, p. 357 6.42, p. 359 Read through p. 370. 17
More on Independence and Disjoint (Mutually Exclusive) Events If I roll two dice, then each die can roll a 3. Neither die influences the other, so they are independent. But since I can roll a 3 on each die simultaneously, they are not disjoint events. +++++++++++++++++++++++++++++++++++++++ A bag contains 3 red balls and 2 green balls. A ball is drawn from the bag, its color is noted, and the ball is set aside. Then a second ball is drawn and its color is noted. Let event A be the event that the first ball is red. Let event B be the event that the second ball is red. Events A and B are not disjoint because both balls can be red. Events A and B are not independent because whether the first ball is red or not alters the probability of the second ball being red. +++++++++++++++++++++++++++++++++++++++ Two events that are disjoint cannot be independent. See last full paragraph on p. 352. 18
6.3 General Probability Models 19
General Addition Rule for Unions of Two Events Rule 4 (Addition Rule). Two events are disjoint (mutually exclusive) if they have no outcomes in common and can never occur simultaneously. If A and B are disjoint, then: P ( A or B) = P( A) + P( B) The above rule does not work if two events are not disjoint (that is, they are not mutually exclusive). Here is the general rule for addition for unions of two events: P( A or B) = P( A) + P( B) P( A and B) 20
Venn Diagrams Venn Diagrams can be very useful in helping find probabilities of unions. 21
Practice Problems 6.46, p. 364 6.47, p. 364 6.52, p. 365 6.51, p. 365 22
More Practice pp. 364-365: 6.48, 6.49, 6.50, 6.53 23
Conditional Probability Probability of an event given, or under the condition that, we know another event. See example 6.18, p. 366 Notation: P( A/ B) or P( B / A) 24
General Multiplication Rule for any Two Events P ( A and B) = P( A) P( B / A) P ( B / A) = P( A and P( A) B) Note: If events A and B are independent, then: P ( B / A) = P( B) 25
Practice Problems 6.54 and 6.55, pp. 369-370 26
HW Problems, pp. 370-371: 6.56, 6.58, 6.60 6.61 (include a Venn diagram for part b) 27
Tree Diagrams Can be used to solve a number of probability problems. Draw a tree diagram to represent this situation: 5% of male high school athletes go on to play college sports. Of this group, 1.7% will go on to play professionally. Of the group of high school athletes which does not play college sports, only 0.01% will play professionally. Question: What s the probability that a high school athlete will play professionally? 28
Tree Diagrams See Examples 6.22-6.23, pp. 372-373 The probability of reaching the end of any complete branch is the product of the probabilities written on each segment. Probabilities after the first level of segments are conditional probabilities. 29
Example 6.23, p. 372 30
Example The weather forecaster at the radio station reports that if it rains on a given day in November, the probability of rain the next day is 70%. If it does not rain on a given day, then the probability of rain the next day is 40%. What is the probability that it will rain on Friday if it is raining on Wednesday? Draw a tree diagram to help solve this problem. 31
Practice Problems 6.70, p. 381 Use a tree diagram to answer (b). Then, use the general multiplication rule to confirm your answer. 6.62 p. 377 6.63, p. 378 6.64, p. 378 Use a tree diagram. 6.67, 6.69 p. 380 6.73, 6.74 p. 381 32
Additional Practice Problems Problems, pp. 383-385: 6.78, 6.79, 6.82, 6.83, 6.84, 6.86 33