Definition of Probability Probability Distribution

1. Presentation Topic
(Probability)

2. Contents
Definition of Probability
The Probability Distribution
a) Discrete Probability Distributions
b) Normal Probability Distributions

3. Definition of Probability
• The term “probability” in Statistics refers to the chances of
occurrence of an event among a large number of possibilities.
OR
• When all the equally possible occurrences have been enumerated,
the probability Pr of an event happening is the ratio of the number
of ways in which the particular event may occur to the total
number of possible occurrences.

4. The Probability Distribution
• Discrete Probability Distributions
• Normal Probability Distributions

5. Discrete Probability
Distributions
• Random Variables
A random variable X represents a numerical value associated with
each outcome of a probability distribution.
A random variable is discrete if it has a finite or countable number of
possible outcome that can be listed.
A random variable is continuous if it has an uncountable number or
possible outcome, represented by the intervals on a number line.

6. Random Variables
Example:
Decide if the random variable X is discrete or continuous.
• The distance your car travels on a tank of gas
The distance your car travels is a continuous random variable because
it is a measurement that cannot be counted.
(All measurements are continuous random variables.)
• The number of students in a statistics class
The number of students is a discrete random variable because it can
be counted.

7. Discrete Probability
Distributions
• A discrete probability distribution lists each possible value the
random variable can assume, together with its probability. A
probability distribution must satisfy the following conditions.
In Words In Symbols
a. The probability of each value of the
discrete random variable is
between 0 and 1, inclusive.
0  P (x)  1
b. The sum of all the probabilities is 1. ΣP (x) = 1

8. Constructing a Discrete
Probability Distribution
Guidelines
Let x be a discrete random variable with possible outcomes x1, x2, …, xn.
• Make a frequency distribution for the possible outcomes.
• Find the sum of the frequencies.
• Find the probability of each possible outcome by dividing its
frequency by the sum of the frequencies.
• Check that each probability is between 0 and 1 and that the
sum is 1.

9. Constructing a Discrete
Probability Distribution
Example:
The spinner below is divided into two sections. The probability of
landing on the 1 is 0.25. The probability of landing on the 2 is 0.75.
Let X be the number the spinner lands on. Construct a probability
distribution for the random variable X.

10. Constructing a Discrete
Probability Distribution
Example:
The spinner below is Spun two times. The probability of landing on
the 1 is 0.25. The probability of landing on the 2 is 0.75. Let X be the
sum of the two spins. Construct a probability distribution for the
random variable X.

11. Constructing a Discrete
Probability Distribution
Example Continued:
2

12. Constructing a Discrete
Probability Distribution
Example Continued:
2

13. Normal Probability
Distributions

14. Properties of Normal
Distributions
• A continuous random variable has an infinite number of possible
values that can be represented by an interval on the number line.
The time spent studying
can be any number
between 0 and 24.
• The probability distribution of a continuous random variable
is called a continuous probability distribution.
Hours spent studying in a day
0 63 9 1512 18 2421

15. Properties of Normal
Distributions
• The most important probability distribution in statistics is
the normal distribution.
x
Normal curve
• A normal distribution is a continuous probability distribution
for a random variable, x. The graph of a normal distribution is
called the normal curve.

16. Properties of Normal
Distributions
i. The mean, median, and mode are equal.
ii. The normal curve is bell-shaped and symmetric about the mean.
iii. The total area under the curve is equal to one.
iv. The normal curve approaches, but never touches the x-axis as it
extends farther and farther away from the mean.
v. Between μ  σ and μ + σ (in the center of the curve), the graph
curves downward. The graph curves upward to the left of μ  σ
and to the right of μ + σ. The points at which the curve changes
from curving upward to curving downward are called the
inflection points.

17. Thanks You