From Randomness to Probability Chapter 14 From Randomness to Probability

Dealing with Random Phenomena 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.

Probability The probability of an event is its long-run relative frequency. While we may not be able to predict a particular individual outcome, we can talk about what will happen in the long run. Each attempt of a random phenomenon is called a trial Example: each time you flip a coin Each trial will generate an outcome which are the individual possibilities Example: each number we see on top when we roll a die.

Probability continued The set of all possible outcomes of an event is the sample space of the event. Example: Rolling a dice S = {1, 2, 3, 4, 5, 6} An event is an outcome (or a set of outcomes) from a sample space. It is usually denoted by a capital letter. The probability of event A is denoted as P(A) Example 1: When flipping three coins, an event may be getting the result THT In this case, the event is one outcome from the sample space. Example 2: When flipping three coins, an event may be getting two tails. In this case, the event is a set of outcomes HTT, THT, TTH from the sample space.

Law of Large Numbers The Law of Large Numbers (LLN) says that the long-run relative frequency of repeated independent events gets closer and closer to the true relative frequency as the number of trials increases. For example, consider flipping a fair coin many, many times. The overall percentage of heads should settle down to about 50% as the number of outcomes increases.

The Law of Large Numbers (cont.) The common (mis)understanding of the LLN is that random phenomena are supposed to compensate some for whatever happened in the past. This is just not true. For example, when 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? 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.

Formal Rules of Probability The probability of any event is between 0 and 1. A probability of 0 indicates that the event will never occur and a probability of 1 indicates that the event will always occur.

Formal Probability (cont.) “Something has to happen rule”: The probability of the set of all possible outcomes of a trial must be 1. P(S) = 1 (S represents the set of all possible outcomes.)

Formal Probability (cont.) Complement Rule: The set of outcomes that are not in the event A is called the complement of A, denoted AC. The probability of an event occurring is 1 minus the probability that it doesn’t occur: P(A) = 1 – P(AC) Example: When flipping two coins, the probability of getting two heads is 0.25 and the probability of not getting two heads is 0.75

Formal probability (cont.) 4) Events A and B are said to be disjoint if they have no outcomes in common. The probability that one or the other occurs is the sum of the probabilities of the two events.

Formal rules (cont.) 4) Disjoint example: Let event A be rolling a die and landing on an even number and event B be rolling a die and landing on an odd number. The outcomes for A are _____________ and the outcomes for B are ______________. These two events are disjoint because they have no outcomes in common. What is the probability?

Formal rules (cont.) 5) The outcome of one trial does not influence or change the outcome of another trial it is said to be independent. Example: When you flip a coin, it does not affect whether the next flip will be heads of tails. If events A and B are independent then the probability of A and B equals the probability of A times the probability of B.

Formal rules (continued) 5) Independent example The probability that the yellow die lands on an even number and the red die lands on an odd number is: