Probability Lesson 4 1 Randomness Probability and Simulation

Probability Lesson 4. 1 Randomness, Probability, and Simulation and Probability with Applications, Statistics and Probability with Applications, 3 rd Edition Starnes &Starnes Tabor & Tabor Bedford Freeman Worth Publishers

Randomness, Probability, and Simulation Learning Targets After you this should lesson, be you should After this lesson, able to: be able to: ü Interpret probability as a long-run relative frequency. ü Dispel common myths about randomness. Statistics and Probability with Applications, 3 rd Edition 2

Dealing with Random Phenomena Chance is all around us. The mathematics of chance behavior is called probability. • When we toss a coin, we know that it will be either a head or a tail, but not which one. • A random phenomenon is a situation in which we know what outcomes could happen, but we don’t know which particular outcome did or will happen. • We use probability to assess the likelihood that a random phenomenon has a particular outcome. • In each case of flipping a coin, the next outcome is uncertain, however, over the long run of many repetitions, a pattern emerges. Statistics and Probability with Applications, 3 rd Edition Slide 14 - 3 3

Randomness, Probability Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. http: //digitalfirst. bfwpub. com/stats_applet_10_ prob. html Probability The probability of any outcome of a chance process is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very large number of repetitions. Statistics and Probability with Applications, 3 rd Edition 4

Randomness, Probability Outcomes that never occur have probability 0. An outcome that happens on every repetition has probability 1. An outcome that happens half the time in a very long series of trials has probability 0. 5. The fact that the proportion of heads in many tosses of a coin eventually closes in on 0. 5 is guaranteed by the law of large numbers. Statistics and Probability with Applications, 3 rd Edition 5

Law of Large Numbers The law of large numbers says that if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches its probability. Statistics and Probability with Applications, 3 rd Edition 6

A MISCONCEPTION – Next, once again consider flipping a fair coin, if heads comes up on each of the first 10 flips, what do you think the chance is that tails will come up on the next flip? • The common misunderstanding of the LLN is that random phenomena are supposed to compensate some for whatever happened in the past. This is just not true!!! Statistics and Probability with Applications, 3 rd Edition Slide 14 - 7 7

The Law of Large Numbers The LLN is important because it "guarantees" stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins It is important to remember that the LLN only applies (as the name indicates) when a large number of observations is considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others Statistics and Probability with Applications, 3 rd Edition 8

Probability • Thanks to the LLN, we know that relative frequencies settle down in the long run, so we can officially give the name probability to that value. BUT REMEMBER: • Probabilities must be between 0 and 1, inclusive. – A probability of 0 indicates impossibility. – A probability of 1 indicates certainty. Statistics and Probability with Applications, 3 rd Edition Slide 14 - 9 9

The Law of Large Numbers Was the moon landing real? According to The Book of Odds, the probability that a randomly selected U. S. adult believes the government staged or faked the Apollo moon landing in July 1969 is 0. 06. • (a) Explain what probability 0. 06 means in this setting. • (b) Does this probability say that if 100 U. S. adults are chosen at random, exactly 6 of them believe the government staged or faked the Apollo moon landing? Explain. Statistics and Probability with Applications, 3 rd Edition Slide 14 - 10 10

Probability • We use probability to assess the likelihood that a random phenomenon has a particular outcome. • If A represents a particular outcome or set of outcomes of a random phenomenon, we write P(A) to denote the probability that event A will occur. There are two requirements for a probability: – A probability is a number between 0 and 1 and can be expressed as fractions, decimals or per cents. – For any event A, 0 ≤ P(A) ≤ 1. ed Statistics and Probability. Larson/Farber with Applications, 3 rd 4 th Edition 11 11

Randomness, Probability, and Simulation Learning Targets After you this should lesson, be you should After this lesson, able to: be able to: ü Interpret probability as a long-run relative frequency. ü Dispel common myths about randomness. Statistics and Probability with Applications, 3 rd Edition 12