Introducing probability

BPS chapter 10

© 2006 W. H. Freeman and CompanyThese PowerPoint files were developed by Brigitte Baldi at the University of California, Irvine and were revised by Ellen Gundlach at Purdue University for the fourth edition.

Objectives (BPS chapter 10)

Introducing probability

The idea of probability

Probability models

Probability rules

Discrete sample space

Continuous sample space

Random variables

Personal probability

Why Do We Need Probability for Statistics?What proportion of all adults bought a lottery ticket in the past 12 months? We don'tknow, but we do have results from the Gallup Poll. Gallup took a random sample of 1523 adults. The poll found that 868 or 57% of the people in the sample bought tickets.

It seems reasonable to use this 57% as an estimate of the unknown proportion in the population. It’s a fact that 57% of the sample bought lottery tickets—we know becauseGallup asked them. We don’t know what percent of all adults bought tickets, but we estimate that about 57% did. This is a basic move in statistics: use a result from a sampleto estimate something about a population. What if Gallup took a second random sample of 1523 adults? The new sample would have different people in it. It is almost certain that there would not be exactly 868 positive responses. That is, Gallup’s estimate of the proportion of adults who bought a lottery ticket will vary from sample to sample.

This is where we need facts about probability to make progress in statistics. Because Gallup uses chance to choose its samples, the laws of probability govern the behavior of the samples. Gallup says that they can say "with 95% confidence that the maximum margin of sampling error is ±3 percentage points." That is, they are 95% confident that an estimate from one of Their samples comes within ±3 percentage points of the truth about the population of all adults.

The first step toward understanding this statement is to understand what “95% confident” means. Our purpose in this chapter is to understand the language of probability, but without going into the mathematics of probability theory.

Meaning of a probabilityWe have several ways of defining a probability, and this has consequences on its intuitive meaning.

Theoretical probability

From understanding the phenomenon and symmetries in the problem

Example: Six-sided fair die: Each side has the same chance of turning up; therefore, each has a probability 1/6.

Example: Genetic laws of inheritance based on meiosis process.

Empirical probability

From our knowledge of numerous similar past events

Mendel discovered the probabilities of inheritance of a given trait from experiments on peas, without knowing about genes or DNA.

Example: Predicting the weather: A 30% chance of rain today means that it rained on 30% of all days with similar atmospheric conditions.

Personal probability

From subjective considerations, typically about unique events

Example: Probability of a large meteorite hitting the Earth. Probability of

life on Mars. These do not make sense in terms of frequency.

A personal probability represents an individual’s personal degree of

belief based on prior knowledge. It is also called Baysian probability for

the mathematician who developed the concept.

We may say “there is a 40% chance of life on Mars.” In fact, either there

is or there isn’t life on Mars. The 40% probability is our degree of belief,

how confident we are about the presence of life on Mars based on what

we know about life requirements, pictures of Mars, and probes we sent.

Our brains effortlessly calculate risks (probabilities) of all sorts, and

businesses try to formalize this process for decision-making.

A phenomenon is random if individual

outcomes are uncertain, but there is

nonetheless a regular distribution of

outcomes in a large number of

repetitions.

Randomness and probability

The probability of any outcome of a random phenomenon can be

defined as the proportion of times the outcome would occur in a very

long series of repetitions.

Coin toss The result of any single coin toss is

random. But the result over many tosses

is predictable, as long as the trials are

independent (i.e., the outcome of a new

coin toss is not influenced by the result of

the previous toss).

The result of any single coin toss is

random. But the result over many tosses

is predictable, as long as the trials are

independent (i.e., the outcome of a new

coin toss is not influenced by the result of

the previous toss).

First series of tossesSecond series

The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials.

The trials are independent only when

you put the coin back each time. It is

called sampling with replacement.

Two events are independent if the probability that one event occurs

on any given trial of an experiment is not affected or changed by the

occurrence of the other event.

When are trials not independent?

Imagine that these coins were spread out so that half were heads up and half

were tails up. Close your eyes and pick one. The probability of it being heads is

0.5. However, if you don’t put it back in the pile, the probability of picking up

another coin and having it be heads is now less than 0.5.

Probability models mathematically describe the outcome of random

processes. They consist of two parts:

1) S = Sample Space: This is a set, or list, of all possible outcomes

of a random process. An event is a subset of the sample space.

2) A probability for each possible event in the sample space S.

Probability models

Example: Probability Model for a Coin Toss

S = {Head, Tail}

Probability of heads = 0.5

Probability of tails = 0.5

Important: It’s the question that determines the sample space.

Sample space

A. A basketball player shoots three free throws. What are the possible sequences of hits (H) and misses (M)?

H

H

H - HHH

M …

M

M - HHM

H - HMH

M - HMM

…

S = {HHH, HHM, HMH, HMM, MHH, MHM, MMH, MMM }

Note: 8 elements, 23

B. A basketball player shoots three free throws. What is the number of baskets made?

S = {0, 1, 2, 3}

Coin Toss Example: S = {Head, Tail}Probability of heads = 0.5Probability of tails = 0.5

1) Probabilities range from 0 (no chance of the event) to1 (the event has to happen).

For any event A, 0 ≤ P(A) ≤ 1

Probability rules

2) The probability of the complete sample space must equal 1.

P(sample space) = 1

P(head) + P(tail) = 0.5 + 0.5 = 1

3) The probability of an event not occurring is 1 minus the probability that does occur.

P(A) = 1 – P(not A)

P(tail) = 1 – P(head) = 0.5

Probability of getting a head = 0.5We write this as: P(head) = 0.5

P(neither head nor tail) = 0P(getting either a head or a tail) = 1

Probability rules (cont'd)

4) Two events A and B are disjoint if they have

no outcomes in common and can never happen

together. The probability that A or B occurs is

the sum of their individual probabilities.

5) P(A or B) = P(A) + P(B) ─ P(A and B)

Example: If you flip two coins and the first flip does not affect the second flip,

S = {HH, HT, TH, TT}. The probability of each of these events is 1/4, or 0.25.

The probability that you obtain “only heads or only tails” is:

P(HH or TT) = 0.25 + 0.25 − 0= 0.50

A and B disjoint

A and B not disjoint

Note: Discrete data contrast with continuous data that can take on any one of

an infinite number of possible values over an interval.

Dice are good examples of finite sample spaces. Finite means that there is a limited number of outcomes.

Throwing 1 die:

S = {1, 2, 3, 4, 5, 6}, and the probability of each event = 1/6.

Discrete sample space

Discrete sample spaces deal with data that can take on only certain

values. These values are often integers or whole numbers.

In some situations, we define an event as a combination of outcomes. In that case, the probabilities need to be calculated from our knowledge of the probabilities of the simpler events.

Example: You toss two dice. What is the probability of the outcomes summing to five?

There are 36 possible outcomes in S, all equally likely (given fair dice).

Thus, the probability of any one of them is 1/36.

P(the roll of two dice sums to 5) =

P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 * 1/36 = 1/9 = 0.111

This is S:

{(1,1), (1,2), (1,3), ……etc.}

The gambling industry relies on probability distributions to calculate the odds of winning. The rewards are then fixed precisely so that, on average, players lose and the house wins.

The industry is very tough on so-called “cheaters” because their probability to win exceeds that of the house. Remember that it is a business, and therefore it has to be profitable.

Give the sample space and probabilities of each event in the following cases:

A couple wants three children. What are the arrangements of boys (B) and girls (G)?

Genetics tells us that the probability that a baby is a boy or a girl is the same, 0.5.

→ Sample space: {BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG}→ All eight outcomes in the sample space are equally likely. → The probability of each is thus 1/8.

A couple wants three children. What are the numbers of girls (X) they could have?

The same genetic laws apply. We can use the probabilities above to calculate the probability for each possible number of girls.

→ Sample space {0, 1, 2, 3} → P(X = 0) = P(BBB) = 1/8 → P(X = 1) = P(BBG or BGB or GBB) = P(BBG) + P(BGB) + P(GBB) =

3/8