Probability



4 - Probability

In this section we examine basic ideas about probabilities:

• what they are and where they come from • simple probability models

• conditional probabilities • independent events

Ex 1: I toss a fair coin (where ‘fair’ means ‘equally likely outcomes’)

• What are the possible outcomes?

• What is the probability it will turn up heads?

Ex 2: I choose a person at random and check which eye she/he winks with

▪ What are the possible outcomes?

▪ What is the probability they wink with their left eye?

WHAT ARE PROBABILITIES?

A probability is a number

WHERE DO PROBABILITIES COME FROM?

Probabilities from models:

The probability of getting a four when a fair dice is rolled is

Probabilities and proportions

e.g. The proportion of U.S. citizens who are left handed is 0.1; a randomly selected U.S. citizen is left handed with a probability of approximately 0.1

Probabilities from data or empirical probabilities:

In a survey conducted by a STAT 110 course, there were 348 WSU students sampled. 212 of these students said they regularly drink alcohol. Based on the survey the probability that a randomly chosen Winona State student drinks alcohol is

Subjective probabilities:

The probability that there will be another outbreak of ebola in Africa within the next year is 0.1.

The probability of rain in the next 24 hours is very high, e.g. 70% chance.

SIMPLE PROBABILITY MODELS AND NOTATION

“The probability that an event A occurs” is written in shorthand as

|For equally likely outcomes, and a given event A: |

| |

| |

Example 0: Roll two die, one red and one white, and consider the sum.

| | White Die |

|Red Die | |

| |1 |2 |3 |4 |5 |6 |

|1 |1,1 |1,2 |1,3 |1,4 |1,5 |1,6 |

|2 |2,1 |2,2 |2,3 |2,4 |2,5 |2,6 |

|3 |3,1 |3,2 |3,3 |3,4 |3,5 |3,6 |

|4 |4,1 |4,2 |4,3 |4,4 |4,5 |4,6 |

|5 |5,1 |5,2 |5,3 |5,4 |5,5 |5,6 |

|6 |6,1 |6,2 |6,3 |6,4 |6,5 |6,6 |

Find the probability of the following events

PROBABILITIES FROM DATA - SOME BASIC IDEAS

Example 1 - In 1996, 6631 in individuals in New Zealand died from coronary heart disease. The numbers

of deaths classified by age and gender are:

Sex

|Age |Male |Female |Total |

|< 45 |79 |13 |92 |

|45 - 64 |772 |216 |988 |

|65 - 74 |1081 |499 |1580 |

|>74 |1795 |2176 |3971 |

|Total |3727 |2904 |6631 |

Find the probability that a randomly chosen member of this population at the time of death was:

a) under 45

b) male assuming that the person was younger than 45.

c) male and was over 64.

d) over 64 given they were female.

CONDITIONAL PROBABILITY and INDEPENDENCE

• The sample space is reduced.

• Key words that indicate conditional probability are: given, amongst, for those with, …

Conditional Probability

|“The probability of event A occurring given that event B has already occurred” |

|is written in shorthand as |

Independence

Example 0 (cont’d): Rolling two die, one white and one red.

Let A = event that the white die > 4

B = event that the red die = 6

Find the following:

P(A|B) =

P(A)

P(B)

P(A and B) =

Example 2 - A study was conducted in 1991 by the University of Wisconsin and the Wisconsin Department of Transportation in which linked police reports and hospital discharge records were used to assess, among other things, the risk for brain injury for motorcyclists in motor-vehicle crashes. The data shown below can be used to examine the relationship between helmet use and whether brain injury was sustained in the accident.

| |Brain Injury |No Brain Injury |Row Totals |

|Helmet Worn |17 |977 |994 |

|No Helmet |97 |1918 |2015 |

|Column Totals |114 |2895 |3009 |

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a) What is the probability that a motorcycle accident victim in Wisconsin suffered brain injury?

b) What is the probability that a motorcyclist involved in an accident was wearing a helmet?

c) What is the probability that a motorcyclist suffered brain injury given that they were wearing a helmet?

d) What is the probability that a motorcyclist not wearing a helmet suffered brain injury?

e) How many times more likely is a motorcyclist not wearing a helmet to sustain a brain injury?

This ratio is called the _____________________________ or _______________________________. .

Building a contingency table from a story

Example 3 - University of Florida sociologist, Michael Radelet, believed that if you killed a white person in Florida the chances of getting the death penalty were three times greater than if you had killed a black person. In a study Radelet classified 326 murderers by race of the victim and type of sentence given to the murderer. 36 of the convicted murderers received the death sentence. Of this group, 30 had murdered a white person whereas 184 of the group that did not receive the death sentence had murdered a white person.

(Gainesville Sun, Oct 20 1986)

Let W be the event that

D be the event that

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

(a) What proportion of the murderers in this study received the death sentence?

b) If a victim from this study was white, what is the probability that the murderer of this victim received the death sentence?

c) If a victim from this study was black, what is the probability that the murderer of this victim received the death sentence?

(d) What is the relative risk (RR) of receiving a death sentence associated with a black murderer

being convicted of murdering a white person?

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Events A and B are said to be independent if

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