Lect 01: ReviewTue 23 Feb 2016

Outline of the Review

Terminology and basic concepts

Problem

• A spinner has 4 equal sectors colored yellow, blue, green and red. • What are the chances of landing on blue after spinning the spinner?

• What are the chances of landing on red?

• Answer: • The chances of landing on blue are 1 in 4, or one fourth.

• The chances of landing on red are 1 in 4, or one fourth.

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Definitions

Definition Example

An experiment is a situation involving

chance or probability that leads to

results called outcomes.

In the problem above, the experiment

is spinning the spinner.

An outcome is the result of a single

trial of an experiment.

The possible outcomes are landing

on yellow, blue, green or red.

An event is one or more outcomes of

an experiment.

One event of this experiment is

landing on blue.

Probability is the measure of how

likely an event is.

The probability of landing on blue is

one fourth.

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Probability Of An

Event

P(A) =The Number Of Ways Event A Can Occur

The total number Of Possible Outcomes

Terminology: an elementary

event (also called an atomic

event or simple event) is an

event which contains only a

single outcome in the sample

space.

Probability, Event and Sample Space

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Ex 2: we have a sample space of sentences and we are interested in the length of these sentences. A relevant event would be the set of all sentences that contain exactly 8 words. And again we can describe this set as the outcome for which the variable "numberOfWords" takes the value 8

Ex 1: we have a sample space consisting of words. An event in that sample space can be the set of NOUNS, ie all the words that belong to the category NOUN. One way of describing this subset is to say that the property PartOfSpeech has the value NOUN. = is an element of

a random variable is a numerical measurement of outcomes

P(X = x)

Formula & Calculations

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Calculations: 6x6x6=216; 26x26x26=17576; 216/17576=0.01228949

if A is an event, and x1 to xn are its individual outcomes, then the probability of A can be computed by summing the probability of each outcome because they are disjoint or mutually exclusive.

Read as: sum from i=1 to nor sum over all the elements of the set

There are 26 ways of choosing the first letter, 26 ways of choosing the 2nd letter and 26 ways of choosing the third letter, ie 26*26*26 = 263

But there are only 6 ways of choosing the first vowel… Since we assume that all strings are equally

possible, the probability is simply 1 over the total number of strings.

In order to get the probability of the 3-vowel string, we can simply add the strings that contain exactly 3 vowels. So 6 to the power of 3 over 26 to the power of 3 gives us approximately.012

https://www.mathsisfun.com/numbers/sigma-calculator.html

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Calculations: 6x6x6=216; 26x26x26=17576; 216/17576=0.01228949

They are equivalent!

Probability of an event

The Number Of Ways Event A Can Occur

The total number Of Possible Outcomes

In one roll of a die, what is the probability of getting an odd number?

Because rolling 1, 3 and 5 are mutually exclusive events:

The number of ways the event A can occur is 3. That is, A [1, 3, 5]. 1 can happen 1/6; 3 can happen 1/6; 5 can happen 1/6 = tot 3/6 = 1/2

Event A has 3 favourable outcomes; We sum over the probability of all the favarouble outcomes in an event

Voted for review: 4%Quiz 1: The lottery(only one answer is correct)

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Solutions:

1. 0.01 - incorrect. The probability of an event and its complement must sum to 1.

2. 0.99 - correct. The complement of A has probability 1 - P(A).

3. Impossible to tell - incorrect. The complement of A must have probability 1 - P(A).

100/100 = 1 the probability that the event will occerr

The probability on winning a lottery is 1/100 = 0.01

The probability of NOT winning a lottery is 100/100 – 1/100 or 1 – 0.01

We know that probability must sum up to 1. Then the complement of the winning event is 1 - 0.01

What is a complement?

Voted for review: 8%Quiz 2: Events in a sample space(more than 1 answers can be correct)

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Solutions

1. P(A or B) < P(A and B) - incorrect. Since the union includes the intersection, it can never have lower probability.

2. 2. P(A or B) = P(A and B) - correct. This is possible as a limiting case, for example, when A = B.

3. 3. P(A or B) > P(A and B) - correct. This holds as soon as there is some outcome with a positive probability in A or B that is not in the intersection.

a.k.a and

a.k.a or

The end