Kenwood Academy
AP Statistics
Chapter 5 Notes – Probability: What are the Chances?
|5.1 Randomness, Probability, and |Objective: |
|Simulation |Assignment 5.1 page 293 #1-21odd, 31-38 all |
| |Interpret probability as a long-run relative frequency in context. |
|The Idea of Probability |Use simulation to model chance behavior. |
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| |Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. |
| |The law of large numbers says that if we observe more and more repetitions of any chance process, the proportion of times that a |
|probability |specific outcome occurs approaches a single value. |
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|Example: |The probability of any outcome of a chance process is a number between 0 (never occurs) and 1(always occurs) that describes the |
| |proportion of times the outcome would occur in a very long series of repetitions. |
|Example: | |
| |Page 284 |
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| |Check Your Understanding page 286 |
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|Myths about Randomness | |
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|Simulation | |
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| |Myth of short-run regularity. See Example page 287-288 |
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| |Myth of the “law of averages” See page 288-289 |
|Performing a Simulation: | |
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| |The imitation of chance behavior, based on a model that accurately reflects the situation, is called a simulation. |
|Examples: |We can use physical devices, random numbers (e.g. Table D), and technology to perform simulations. |
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|Example: | |
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| |State: What is the question of interest about some chance process? |
| |Plan: Describe how to use a chance device to imitate one repetition of the process. Explain clearly how to identify the outcomes |
| |of the chance process and what variable to measure. |
| |Do: Perform many repetitions of the simulation. |
|5.2 Probability Rules |Conclude: Use the results of your simulation to answer the question of interest. |
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| |Page 290 and 291 |
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| |Check Your Understanding page 292 |
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|Probability Models | |
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|Random | |
| |Objective: |
|Probability |Describe a probability model for a chance process. |
| |Use basic probability rules, including the complement rule and the addition rule for mutually exclusive events. |
| |Use a Venn diagram to model a chance process involving two events. |
| |Determine probabilities form a two-way table. |
| |Use the general addition rule to calculate P(A ∪ B) |
| |Find the probability that an event occurs using a two-way table. |
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| |Probability is the branch of mathematics that describes the pattern of chance outcomes. Chance behavior is unpredictable in the |
| |short run but has a regular and predictable pattern in the long run. |
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|Example |– A random event is one whose outcome we cannot predict beforehand. |
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| |– the proportion of times an outcome would occur in a very long series of repetitions. Probability is long-term relative |
| |frequency. |
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| |The sample space S of a random phenomenon is the set of all possible outcomes. |
| |An event is any outcome or set of outcomes of a random phenomenon. The event is a subset of the sample space. |
| |A probability model is a mathematical description of a random phenomenon consisting of two parts: a sample space S and a way of |
| |assigning probabilities to events. |
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| |[pic] |
| |Give a probability model for the chance process of rolling two fair, six-sided dice |
| |Sample Space for rolling 2 dice and counting the pips: |
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| |S = |
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| |Suppose event A is defined as “sum is 5.” |
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| |A = |
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| |What is P(A)? P(A) = |
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|Probability Rules |Suppose event B is defined as “sum is not 5.” What is P(B)? |
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| |Rule 1 |
| |Any probability is a number between 0 and 1. |
| |0 ( P(A) ( 1 for any event A. |
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| |Rule 2 |
| |The sum of all possible outcomes must equal 1 |
| |P(S) = 1 |
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| |Rule 3 |
| |If all outcomes in the sample space are equally likely, the probability that event A occurs can be found using the formula |
| |[pic] |
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| |Rule 4 |
| |The probability that an event does not occur is 1 minus the probability that the event does occur. This is called the complement|
| |rule. |
| |Complement rule: P(AC) = 1 – P(A) |
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| |Rule 5 |
| |Two events A and B are disjoint (mutually exclusive) if they have no outcomes in common and so can never occur simultaneously. |
| |If two events are disjoint, the probability that one or the other occurs is the sum of their individual probabilities. This is |
| |called the Addition rule |
| |Addition rule for mutually exclusive events: If A and B are mutually exclusive, |
| |P(A or B) = P(A) + P(B). |
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| |Rule 5 |
| |If two events are not mutually exclusive, the probability that one or the other occurs is the sum of their individual |
| |probabilities minus the probability of both events occurring. This is called the General Addition rule |
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|Example: |General Addition Rule: If A and B are any two events resulting from some chance process, then |
| |P(A or B) = P(A) + P(B) – P(A and B) |
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| |Rule 6 |
| |Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. This is |
| |called the multiplication rule for independent events. |
| |If A and B are independent, |
| |P(A and B) = P(A)P(B) |
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| |Check Your Understanding page 303 |
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|Two-Way Tables and Probability | |
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|Example | |
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| |When finding probabilities involving two events, a two-way table can display the sample space in a way that makes probability |
| |calculations easier. |
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| |Page 303 and #50 page 310 |
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| |Note that the example on page 303 illustrates the fact that we can’t use the addition rule for mutually exclusive events unless |
| |the events have no outcomes in common. |
| |The Venn diagram below illustrates why. |
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|Example | |
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|Venn Diagrams and Probability |The General Addition Rule: |
| |If A and B are any two events resulting from some chance process, then |
| |P(A or B) = P(A) + P(B) – P(A and B) |
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| |Check Your Understanding page 305 |
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| |The complement AC contains exactly the outcomes that are not in A. |
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|Example: | |
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| |The events A and B are mutually exclusive (disjoint) because they do not overlap. That is, they have no outcomes in common. |
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| |The intersection of events A and B (A ∩ B) is the set of all outcomes in both events A and B. |
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| |[pic] |
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| |Remember ∪ for union and ∩ for intersection |
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|5.3 Conditional Probability and | |
|Independence | |
| |#54 and 56 page 311 |
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|Conditional Probability | |
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|Examples: | |
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| |Objectives: |
| |When appropriate, use a tree diagram to describe chance behavior. |
| |Use the general multiplication rule to solve probability questions. |
| |Determine whether two events are independent. |
| |Find the probability that an event occurs using a two-way table. |
|Conditional Probability and |When appropriate, use the multiplication rule for independent events to compute probabilities. |
|Independence |Compute conditional probabilities. |
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| |The probability we assign to an event can change if we know that some other event has occurred. This idea is the key to many |
| |applications of probability. |
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| |Definition: The probability that one event happens given that another event is already known to have happened is called a |
| |conditional probability. Suppose we know that event A has happened. Then the probability that event B happens given that event A |
| |has happened is denoted by P(B | A). |
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| |Page 312-313 (Read) |
| |Check Your Understanding 314 |
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|Example: | |
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| |When knowledge that one event has happened does not change the likelihood that another event will happen, we say the two events |
| |are independent. |
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| |Definition: Two events A and B are independent if the occurrence of one event has no effect on the chance that the other event |
| |will happen. In other words, events A and B are independent if |
| |P(A | B) = P(A) and P(B | A) = P(B). |
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|Tree Diagrams | |
| |[pic] |
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| |Check Your Understanding page 317 |
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|Replacement | |
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|General Multiplication Rule | |
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| |What are the possible outcomes of flipping a coin and rolling a die? What is the probability of each outcome? |
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|Example: | |
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| |Multiplication Principle – If you can do one task in a number of ways and a second task in b number of ways, then both tasks can |
| |be done in |
| |a ( b number of ways. |
| |What are the number of outcomes of flipping a coin and rolling a die? |
| |2 ( 6 = 12 |
| |(2 ways the coin comes up)(6 ways the die comes up) = 12 outcomes in the sample space |
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| |Sampling with replacement |
| |Sampling without replacement |
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| |How many three digit numbers can you make? |
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| |How many three digit numbers can you make without replacement? |
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|Independence: A Special | |
|Multiplication Rule | |
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|Example: | |
| |The idea of multiplying along the branches in a tree diagram leads to a general method for finding the probability P(A ∩ B) that |
| |two events happen together. |
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| |The probability that events A and B both occur can be found using the general multiplication rule |
| |P(A ∩ B) = P(A) • P(B | A) |
| |where P(B | A) is the conditional probability that event B occurs given that event A has already occurred. |
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|Calculating Conditional | |
|Probabilities |Example 319 (Read) |
| |Example 320 (Read and Calculate) |
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|Example: | |
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| |Check Your Understanding page 321 |
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| |Definition: Multiplication rule for independent events |
| |If A and B are independent events, then the probability that A and B both occur is |
| |P(A ∩ B) = P(A) • P(B) |
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| |Check Your Understanding page 323 |
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| |If we rearrange the terms in the general multiplication rule, we can get a formula for the conditional probability P(B | A). |
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| |General Multiplication Rule Conditional Probability Formula |
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| |P(A ∩ B) = P(A) • P(B | A) ( |
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| |Example page 325 |
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| |#98 page 332 |
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