Constructing Probability Distributions: Statistics and Probability Lesson, Slides of Mathematics

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CONSTRUCTING

PROBABILITY

DISTRIBUTIONS

STATISTICS AND

PROBABILITY

Lesson Objectives

At the end of this lesson, you are expected to:

illustrate a probability distribution for a discrete

random variable and its properties;

compute probabilities corresponding to a given

random variable; and

 construct the probability mass function of a

discrete random variable and its corresponding

histogram.

Pre-Assessment A

Recap:

A. Find the probability of the following events.

Sampl

e

Space

Event

(E)

Probabilit

y

P(E)

1. Getting an even number in a single roll of a

die

2. Getting a sum of 6 when two dice are rolled
3. Getting a sum of 11 when two dice are rolled
4. Getting a doubles when two dice are rolled
5. Getting an odd number and a tail when a die

is rolled, and a coin is tossed simultaneously

Two dice

6 3 3/6 or

Pre-Assessment A

Recap:

A. Find the probability of the following events.

Sampl

e

Space

Event

(E)

Probabilit

y

P(E)

1. Getting an even number in a single roll of a

die

2. Getting a sum of 6 when two dice are rolled
3. Getting a sum of 11 when two dice are rolled
4. Getting a doubles when two dice are rolled
5. Getting an odd number and a tail when a die

is rolled, and a coin is tossed simultaneously

Two dice

6 3 3/6 or

36 2 2 /36^ or

Pre-Assessment A

Recap:

A. Find the probability of the following events.

Sampl

e

Space

Event

(E)

Probabilit

y

P(E)

1. Getting an even number in a single roll of a

die

2. Getting a sum of 6 when two dice are rolled
3. Getting a sum of 11 when two dice are rolled
4. Getting a doubles when two dice are rolled
5. Getting an odd number and a tail when a die

is rolled, and a coin is tossed simultaneouslyA die and a coin

6 3 3/6 or

36 2 2/36^ or

1/

36 6 6/36^ or

1/

1H, 2H, 3H, 4H, 5H, 6H,

1T, 2T, 3T, 4T, 5T, 6T

3 /12 or

Pre-Assessment A

Recap:

A. Find the probability of the following events.

Sample

Space

Event

(E)

Probability

P(E)

6. Getting an ace when a card is drawn

from a deck of cards

7. Getting a black card or a 10 when a

card is drawn from a deck of cards

8. Getting a red queen when card is drawn

from a deck of cards

52 4 4/52 or 1/

Pre-Assessment A

Recap:

A. Find the probability of the following events.

Sample

Space

Event

(E)

Probability

P(E)

6. Getting an ace when a card is drawn

from a deck of cards

7. Getting a black card or a 10 when a

card is drawn from a deck of cards

8. Getting a red queen when card is drawn

from a deck of cards

(^52 4) 4/52 or 1/

(^28) 28/52 or 7/

52 2 2/52 or 1/

Pre-Assessment B

Recap:

Evaluating an algebraic expression means finding the value of an

expression when the variables take on certain values.

B. For the given value of x, evaluate P(x).

X

P(x) = P(x) =

P(x) = P(x) =

P(x) =

P(x) = P(x) =

P(x) = P(x) = or P(x) =^ P(x) =

P(x) = or P(x) =

P(x) = or 2 P(x) =

P(x) = or 2 P(x) =

Discussion

A discrete probability distribution or a

probability mass function consists of

the values a random variable can assume

and the corresponding probabilities of the

values.

Discussion

Illustrative Example: Finding the probability

corresponding to a given random variable

Number of Tails

Suppose three coins are tossed. Let Y be the

random variable representing the number of tails

that occur. Find the probability of each of the values

of the random variable Y.

Discussion Points

Step 2:

Count the number of

tails in each outcome

in the sample space

and assign this

number to this

outcome.

Discussion

Step 3:

There are four possible values of

the random variable Y

representing the number of tails.

These are 0, 1, 2, and 3. Assign

probability values P(Y) to each

value of the random variable.

The Probability Distribution or the Probability

Mass Function of Discrete Random Variable Y

Number of

tails

Sample Space Event (y) Probability

P(y)

Example 1

Number of Blue Balls

Two balls are drawn in succession without

replacement from an urn containing 5 red balls

and 6 blue balls. Let Z be the random variable

representing the number of blue balls. Construct

the probability distribution of the random

variable Z.