5.1-5.3 Notes from MyLabsPlus textbook PDF

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Summary

Professor Shkembi
Boxplots
Chapter 5.1: Random, Probability, Theoretical Probability/Empirical Probability, Simulations
Chapter 5.2: Finding Theoretical Probabilities, Complement, Equally Likely Outcomes, Sample Space, Event, Mutually Exclusive Events...

Description

The Boxplot - It is a graphic summary. - It provides a visual display of numerical summaries of a distribution of numerical data. - HOW? The box stretches from the first quartile to the third quartile, and a vertical line indicates the median. Whiskers extend to the largest and smallest values that are not potential outliers, and potential outliers are indicated with special marks. - HOW IS IT USED? Boxplots are useful for comparing distributions of different groups of data.

5.1 ●

Random- no predictable pattern occurs and that no digit is more likely to appear than any other. ● Probability- used to measure how often random events occur. ○ Probabilities are defined as relative frequency. Theoretical and Empirical Probability Theoretical Probability- the relative frequency at which an event happens after infinitely  many repetitions. - Theoretical Probabilities are long-run relative frequencies. - Example: a coin has 50% probability of coming up heads, we mean that if it were possible to flip the coin infinitely m  any times, then exactly 50% of the flips would be heads. - “What should happen.” - Theoretical Probabilities are NOT based off experiments. Empirical Probability- the relative frequency based on an e  xperiment or on observations of a real-life process. 6 or 60%. Example: My empirical probability of getting heads is therefore 10 KEYPOINT: E  mpirical probabilities tell us how often an event occurred in an actual set of experiments or observations. Theoretical probabilities are based on theory and tell us how many times an event would occur is an experiment were repeated infinitely many times. Why do we need both? Simulations- experiments used to produce Empirical probabilities, because the investigators hope that those experiments simulate the situation they are examining.

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5.2 Finding Theoretical Probabilities - Probability are always numbers between 0 and 1. - 0 means that the event never  happens. - 1 means that the event always  happens. - 1- means that the event will not  happen. Complement- “not event”

Equally Likely Outcomes- in some situations, all of the possible outcomes of a random experiment occur with the same frequency. - Example: when you flip a coin, heads and tails are equally likely; when you roll a die, 1,2,3,4,5, and 6 are all equally likely. - When dealing with equally likely outcomes, it is sometimes helpful to list all of the possible outcome. Sample space- a list that contains all possible (and equally likely) outcomes. - Represent the sample space with the letter s. Event- any collection of outcomes in the sample space. - Example: the sample space, s , for rolling a die is the numbers, 1,2,3,4,5, and 6. The event “get an even number” consist of the even outcomes in the sample space, s:  2,4, and 6. - Events are represented by uppercase letters: A,B,C, and so on … Rule 1: A probability is always a number from 0 to 1 or (0% to 100%) inclusive (which means 0 and 1 are allowed). It may be expressed as a fraction, a decimal, or a percent. 0 ≤ P(A) ≤ 1 Rule 2: The probability that an event will not occur is 1 minus the probability that the event will occur. P(A does not occur) = 1 - P(A does occur) c P(A ) = 1 - P(A) c The symbol A is used to represent the complement of A.

N umber of outcomes

Rule 3: Probability of A = P(A) = N umber of all possible outcomes ● This is true only for e  qually likely outcomes. Ex 12: 10 Dice in a Bowl Bowl contains: - 5 red dice (5/10 or 50%) - 3 green dice (3/10 or 30%) - 2 white dice (2/10 or 20%) The words AND and O  R can be used to combine events into new, more complex events. Venn Diagram ● The areas of the regions in Venn diagrams have no numerical meaning. A large area does not contain more outcomes than a small area. ● The rectangle represents the sample space, which consists of all possible outcomes, ● The ovals represent events.

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AND- creates a new event out of two other events. Using OR to Combine Events ● In a Venn Diagram OR events are represented by shading all relevant events. ● Inclusive OR- statistic way of saying O  R ★ AND- fewest ★ OR - most KEYPOINT: T  he word OR creates a new event out of the events A and B. The new event consists of all outcomes that are only in A, that are only in B, and are in both. - Example: In a Venn Diagram, we would count people who wear glasses (Event A), people who wear a hat (Event B), and people who wear both glasses and a hat (Event A AND B). Mutually Exclusive Events- when two events have no outcomes in common - that is, when it is impossible for both events to happen at once. - Example: The events “person is married” and “person is single” are mutually exclusive. - This means that the probability that both events occur at the same time is 0  . Rule 4: The probability that event A happens OR event B happens is: ALWAYS: P (A OR B) = P(A) + P(B) - P(A AND B) Rule 4a: If A and B are mutually exclusive events, the probability that Event A happens OR Event B happens is the sum of the probability that A happens and the probability that B happens. Only if A and B are mutually exclusive Addition Rule: P(A OR B) = P(A) + P(B) ★ If 2 events, A and Bm are not m  utually exclusive, then it is possible for both to happen at once. Ex 8: Rolling a six-sided die - P(Even OR  Greater than 4) = P(Even) +   P(Greater than 4) - P(Even and Greater than 4) - 3/6 + 2  /6 = ⅙

5.3 Conditional Probabilities- focus on just one group of objects and imagine taking a random sample from that group alone. KEYPOINT: In the study of conditional probabilities, P(A | B) means to find the probability that Event A occurs, but to restrict your consideration to those outcomes of A that occur within Event B. It means “the probability of A occurring, given that Event B has occurred.” Formula for Calculating Conditional Probabilities is: Rule 5a: P(A | B) =

P (A AND B) P (B )

Rule 5b: P(A AND  B) = P(B) P(A | B) and also P(A AND B) = P(A) P(B | A) ★ The line in the middle of Event A and Event B is not a division sign. It’s pronounced as “given that.” Probability Formula: P  (A) = ● ●

N(A) N(S )

Associated means “with.” Independent Events- we call variables or events that are not associated. ★ When Events A and B are said to be independent, knowledge that Event B occurred does not change the probability of Event A occurring. ★ If these probabilities are equal, then the two events are independent. If they are not equal, the two events are associated. ○ If the Event A and B are independent, to find the probability of Event A AND B, multiply the probability of A and the probability of B. Formula for Independent Events: P(A | B) = P(A) Multiplication Rule Rule 5c: P(A AND  B) = P(A) P(B)...