INTRODUCTION TO STATISTICS & PROBABILITY

Chapter 4:

Probability: The Study of Randomness

Dr. Nahid Sultana

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Chapter 4 Probability: The Study of Randomness

4.1 Randomness

4.2 Probability Models

4.3 Random Variables

4.4 Means and Variances of Random Variables

4.5 General Probability Rules*

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4.1 Randomness

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The Language of Probability

Toss a coin, or choose an SRS. The result can’t be predicted in advance (i.e. it’s uncertain), because the result will vary when you toss the coin or choose the sample repeatedly.

But there is however a regular pattern in the results, a pattern that comes out clearly only after many repetitions. Definition: We call a phenomenon random if individual outcomes are uncertain but there is however a regular distribution of outcomes in a large number of repetitions.

This remarkable fact is the basis for the idea of probability.

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The Language of Probability (Cont..)

Definition: The probability of any outcome of a random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions.

Concept of Probability

1. Repeat an experiment (or observe a random phenomenon) a

large number of times.

2. Record the number of times a desirable outcome occurs.

3. Compute the ratio: # of times the desirable outcome occurs

Total # of times the experiment was performed

5 Example: The statistics of a particular basketball player state that he makes 4 out of 5 free-throw attempts.

The basketball player is just about to attempt a free throw. What do you estimate the probability that the player makes this next free throw to be?

A. 0.16

B. 50-50. Either he makes it or he doesn’t.

C. 0.80

D. 1.2

Answer: C

The Language of Probability (Cont..)

4.2 Probability Models

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Probability models

Probability Rules

Assigning Probabilities

Independence and the Multiplication Rule

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Probability Models

Descriptions of chance behavior contain two parts: 1. List of possible outcomes and 2. A probability for each outcome.

The set of all possible outcomes of a statistical experiment is called the sample space and it is represented by the symbol S.

An event is an outcome or a set of outcomes of a random phenomenon. That is, an event is a subset of the sample space.

A probability model is a description of some chance process that consists of two parts: a sample space S and a probability for each outcome.

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Probability Models (Cont…)

Example: Give a probability model for the chance process of tossing of a coin.

Sample space , S = {Head, Tail} Each of these outcomes has probability 1/2

Outcome Heads Tails Probability 1/2 1/2

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Example: Give a probability model for the chance process of tossing of a Die.

Each of these outcomes has probability 1/6

Probability Models (Cont…)

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Probability Models (Cont…)

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Sample Space 36 Outcomes Each outcome has probability 1/36.

Example: Give a probability model for the chance process of rolling two fair, six-sided dice―one that’s red and one that’s green.

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Probability Rules

Rule 1. The probability of each outcome is a number between 0 and 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1

Rule 2. The sum of the probabilities of all the outcomes in a sample space equals 1.

If S is the sample space in a probability model, then P(S) = 1.

Rule 3. If two events A and B are mutually exclusive or disjoint

( A ∩ B = Φ, i.e., if A and B have no element in common) , then P(A or B) = P(A U B ) = P(A) + P(B).

This is the addition rule for disjoint events.

Rule 4: The complement of any event A is the event that consists of all the outcomes not in A, written AC.

P(AC) = 1 – P(A)

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Probability Rules (Cont…) Example:

If you draw an M&M candy at random from a bag of the candies, the candy you draw will have one of six colors. The probability of drawing each color depends on the proportion of each color among all candies made. Assume the table below gives the probabilities for the color of a randomly chosen M&M:

Color Brown Red Yellow Green Orange Blue Probability 0.3 0.3 ? 0.1 0.1 0.1

What is the probability of drawing a yellow candy? Answer. 0.1 What is the probability of not drawing a red candy? Answer: 1-0.3 = 0.7

What is the probability that you draw neither a brown nor a green candy? Answer: 1-(0.3+0.1) = 0.6

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Probability Rules (Cont…) Example:

Distance-learning courses are rapidly gaining popularity among college students. Randomly select an undergraduate student who is taking distance-learning courses for credit and record the student’s age. Here is the probability model:

Age group (yr): 18 to 23 24 to 29 30 to 39 40 or over Probability: 0.57 0.17 0.14 0.12

(a)Show that this is a legitimate probability model.

(b)Find the probability that the chosen student is not in the traditional college age group (18 to 23 years).

Each probability is between 0 and 1 and 0.57 + 0.17 + 0.14 + 0.12 = 1

P(not 18 to 23 years) = 1 – P(18 to 23 years) = 1 – 0.57 = 0.43

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Venn Diagrams

Sometimes it is helpful to draw a picture to display relations among several events. A picture that shows the sample space S as a rectangular area and events as areas within S is called a Venn diagram.

Two disjoint events: Two events that are not disjoint, and the event {A and B} consisting of the outcomes they have in common:

A probability model with a finite sample space is called finite.

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Multiplication Rule for Independent Events

If two events A and B do not influence each other, and if knowledge about one does not change the probability of the other, the events are said to be independent of each other.

Multiplication Rule for Independent Events If A and B are independent:

P(A and B) = P(A) × P(B)