UNIT 5 ~ Probability: What Are the Chances? 1 6.1: Simulation Simulation: The of chance behavior, based on a that accurately reflects the phenomenon under consideration. (ex 1) Suppose we are interested in estimating the likelihood of a couple s having a girl among their first four children. Let a flip of a coin represent a birth, with heads corresponding to a girl and tails corresponding to a boy. Assume that girls and boys are equally likely to occur on any birth and that your coin is fair. Simulation in 5 Steps: Step 1: State the problem or describe the random phenomenon. Step 2: State the assumptions. Step 3: Select a model to generate the outcomes. Step 4: Simulate many repetitions. TH TTH TH TH TTTH TH H TTH TTH TH H H TH TTTT Step 5: Summarize the information and state your conclusions. Note: the actual probability of having a girl in this situation is 0.938. Pretty close! In Step 4, we emphasize the importance of,that is that any previous outcome in affects the subsequent outcomes. We must ALWAYS independence during a.
UNIT 5 ~ Probability: What Are the Chances? 2 Other ideas for simulations besides a coin toss??? (ex 2) Suppose a cereal company places one of four toys in every box of cereal. Furthermore, the company claims that each toy is produced in the same quantity so each of the toys is equally likely to show up in a randomly chosen box. Suppose I want to get all 4 toys, but it takes me 15 boxes to find each of the four. Is this evidence that the toys are NOT uniformly distributed? Design a simulation using a die to estimate the probability that it takes 15 or more boxes to get all 4 toys. Then carry out the simulation. Step 1: State the problem or describe the random phenomenon. Step 2: State the assumptions. Step 3: Select a model to generate the outcomes. Step 4: Simulate many repetitions. Step 5: Summarize the information and state your conclusions.
6.2: Probability Models UNIT 5 ~ Probability: What Are the Chances? 3 FOCUS: When we do an experiment or observational study, we would like to know if our results are statistically significant. That is, we want to know if the results we obtained were likely to occur simply by chance. To determine if our results are statistically significant, we need calculate a probability, which is what we will study in the next chapter. The study of is the systematic study of. Three ways to get PROBABILITY: 1. Theoretical Probability: Assumes events and can be found by: P(E)= number of times E occurs number of outcomes in sample space (where P(E) is the probability of a given event occurring) 2. Relative Frequency Probability: As an experiment is again and again, the of the event tends to approach the actual probability. 3. Subjective Probability (Expert Opinion): A probability determined by someone s (an experts). Used in situations where you can not get the probability in any other way like in who till win a football game (no theoretical, can t play multiple times). So the probability is based on information available past performance, etc. NOTE: PROBABILITY is a concept we think about as a long-run relative frequency of a random event. That is, after many many many repetitions, what proportions of outcomes will you find? PROBABILITY VOCABULARY Def: Chance Experiment: is any or in which there is about which of two or more possible outcomes will result flipping a coin rolling a die choosing a card asking a survey question giving a treatment in an experiment Def: Sample Space: The set of all outcomes of a chance experiment. flipping a coin: heads, tails rolling a die: 1, 2,... 6 choosing a card: ace of hearts, 2 of hearts,...king of spades (52 total) Def: Event: Any or set of outcomes of an experiment.
UNIT 5 ~ Probability: What Are the Chances? 4 Def: Random: An event is if individual outcomes are but there is nonetheless a regular of the outcomes in a number of. Def: Probability Model: A mathematical of a random phenomenon (experiment) consisting of two parts: a and a way of assigning to events. (ex) A randomly selected statistics student will be asked for his or her gender, grade, and period. One possible outcome would be: How many possible outcomes are there? (List them) Tree Diagrams: To identify all possible outcomes, we can use a, with each set of branches corresponding to one variable and each end representing one outcome. Sampling With and Without Replacement: Sampling is the process of selecting an object from a large group of objects and inspecting it, noting some feature(s). The object is either put back (sampling ) or put to one side (sampling ).
UNIT 5 ~ Probability: What Are the Chances? 5 (ex) consider a box containing 3 red, 2 blue, and 1 yellow marble. Suppose you wish to sample two marbles: By replacement of the first before the second is drawn. By not replacing the first before the second is drawn. Examine how the tree diagrams differ: With Replacement: Without Replacement: What is P(two reds)? With Replacement: Without Replacement: Multiplication Principle: Another way to determine the total is by using the Multiplication Rule. If you can do one task in n 1 number of ways and a second task in n 2 number of ways, then both tasks can be done in n 1 x n 2 number of ways. (ex) A randomly selected statistics student will be asked for his or her gender, grade, and period. How many possible outcomes are there?
UNIT 5 ~ Probability: What Are the Chances? 6 (ex) Rolling a die or selecting a card from a deck are chance experiments that have equally likely outcomes. Find the following probabilities: P(rolling a 6) = P(rolling an even #) = P(drawing the queen of spades) = P(drawing a queen) = Forming New Events: Let A and B denote 2 events: 1. The event consists of all experimental outcomes that are not in event A. This event is often c called the of A and is usually denoted A', A, ~ A, or A 2. The event consists of all experimental outcomes that are in at least one of the two events, that is: in A, in B, or in both A and B. This event is called the of events A and B and is denoted A B 3. The event consists of all experimental outcomes that are in BOTH of the events A and B. This event is called the of events A and B and is denoted A B
UNIT 5 ~ Probability: What Are the Chances? 7 Def: Two events that have no outcomes in common are said to be or. The of two disjoint events, say A and B, is called the which can be written as. (ex) Suppose we select a random sample of high school students and recorded each student s gender and handedness (right handed or left handed). Let event A = the student is a male and let event B = the student is right handed. How many possible outcomes are there? Draw a tree diagram: List all possible outcomes in the following events: A c c B = c A = c c c ( ( ) ) A B= same as A B A B= A A = PROBABILITY RULES: Rule #1. For any event E, 0 P(E) 1. If P(E) = 1, the event is guaranteed. If P(E) = 0, the event will never occur. Rule #2. If S is the sample space for an experiment, then P(S) = 1. Since S represents all the outcomes in an experiment, one of them has to happen.
UNIT 5 ~ Probability: What Are the Chances? 8 Rule #3. If two events E and F are disjoint (mutually exclusive), then P(E or F) = P( E F) when rolling one die: P(1 or 2) = = P(E) + P(F). P(1 or odd) This rule works for more than 2 disjoint events as well. Rule #4. The complement rule says that for any event E: P(E) + P(not E) = 1 P(E) = 1 - P(not E) If P(rain) = 30%, the P(no rain) = Def: A PROBABILITY DISTRIBUTION is a list of all the in the and their. (ex) Complete the probability distribution and use it to help you answer the following questions. P(F) = P(C or better) = P(A ) = Grade A B C D F Probability.1.3.4.15 INDEPENDENCE: Def: Two events E and F are if: P(E F) = P(E) In other words, the event E is INDEPENDENT if knowing that F occurred does not change the probability of E occurring.
UNIT 5 ~ Probability: What Are the Chances? 9 Independence is NOT the same as being disjoint. If two events are disjoint then they cannot be independent. The concept of independence will be very important to us the rest of the year! Def: Two events are if they are not independent. PROBABILITY RULES (continued): Rule #5. The events E and F are independent if and only if P( E F) = P( E) P( F) Note: this is the for independent events. (ex) Find the probability if you were to toss a coin and roll a die. P(heads and 6) = (ex) The multiplication rule also works for 3 or more events. a. P(heads and 6 and Queen) = b. P(5 heads in a row) = c. P(at least one tail in 5 flips) = 1 - P(5 heads in a row) = 6.3: General Probability Rules: SUMMARY OF 6.2 PROBABILITY RULES Rule #1. 0 P(E) 1 for any event A. Rule #2. P(S) = 1 Rule #3. Addition Rule: If A and B are disjoint events, then P A B = P(A) + P(B) P(A or B) = ( ) Rule #4. Complement Rule: For any event A, P(A )=1-P(A) Rule #5. Multiplication Rule: If A and B are independent events, then P(A and B)= P( E F) = P( E) P( F)
UNIT 5 ~ Probability: What Are the Chances? 10 GENERAL ADDITION RULE FOR THE UNION OF ANY TWO EVENTS: In general, for ANY two events A and B: P( A B) = P(A) + P(B) - P( A B) Venn Diagram:. Note: if the events were disjoint, then P( A B) = 0 (ex) When drawing a card from a deck, let A = the card is a heart and B = the card is a queen. Find the probability of drawing a heart or queen. GENERAL MULTIPLICATION RULE FOR THE INTERSECTION OF ANY TWO EVENTS: In general, for ANY two events A and B, P( A B) P( A) P( B A) =. Here, P( B A ) is the that B occurs, given the information that A occurs. (ex) If you draw 2 cards from a deck without replacement and let event A = 1st card is a heart and event P A B. B = 2 nd card is black. Find ( ) BAYES RULE: When P(A) > 0, the conditional probability of B, given A is: P( B A ) = CONDITIONAL PROBABILITY & INDEPENDENCE: Two events A and B are if: P( B A ) = P(B) (ex) Suppose that at a high school, 40% of the students are upperclassmen, 70% of the upperclassmen have a drivers license while only 10% of the lowerclassmen have one. Express this information in symbolic form and in a tree diagram. Let U = student is an upperclassman and D = student has a drivers license. Symbolic Form:
UNIT 5 ~ Probability: What Are the Chances? 11 Tree Diagram: If you select a student at random, what is the probability: a. he is an upperclassman and has a drivers license? ( D) P U = b. he has a drivers license? P(D) = P(D U) + P(D U ) = c. he is an upperclassman given that he has a drivers license? P(U D) = P( U D) P( D) = d. he is an upperclassman or has a drivers license? P(U or D) = P(U) + P(D) - P( U D) = e. Are U and D disjoint events? Explain. f. Are U and D independent events? Explain. (ex) In a certain city, 45% of registered voters are Republicans, 40% are Democrats and the rest are Independents. Also, 40% of the Republicans are women, 55% of the Democrats are women and 60% of the Independents are women.
UNIT 5 ~ Probability: What Are the Chances? 12 Make a tree diagram to display this information. Suppose you randomly selected one registered voter. What is the probability that you choose: a. person who is Republican and a woman b. person who is a woman c. person who is male or an Independent d. person who is a Republican if you know he is a male Are being a woman and being a Democrat independent? Justify.
UNIT 5 ~ Probability: What Are the Chances? 13 (ex) A survey of 270 students was conducted to investigate ice cream flavor preference. It is found that 64 students like strawberry, 94 like vanilla, 58 like chocolate, 26 like both strawberry and vanilla, 28 like both strawberry and chocolate, 22 like both vanilla and chocolate, and 14 like all three flavors. How many of the 270 students do not like any flavor?