The normal probability curve is a bell-shaped curve that is used to represent probability distributions of many random variables. Some key properties of the normal curve are: 1) Approximately 68% of values fall within one standard deviation of the mean, 95% within two standard deviations, and 99% within three standard deviations. 2) The curve is perfectly symmetrical, with the mean, median, and mode all being the same value. 3) It approaches but never touches the x-axis and theoretically extends from negative to positive infinity. Standard deviation is used to measure deviations from the mean.

1. 1
Probability and the
Normal Curve

2. Probability Definition in Math
- means possibility.
Probability is a measure of the likelihood of an event to
occur.
It is a branch of mathematics that deals with the occurrence
of a random event. The value is expressed from zero to one.
The meaning of probability is basically the extent to which
something is likely to happen.

3. Probability
- It is quantified as a number between 0 and 1 (where 0
indicates impossibility and 1 indicates certainty)
- The probability of all the events in a sample space adds up
to 1.
- A simple example is the tossing of an unbiased coin. Since
the coin is unbiased, the two outcomes (head or tails) are
equally probable. Since no other outcome is possible, the
probability is ½ or 50% of either ‘head’ or ‘tail’.
- when we toss a coin, either we get Head OR Tail, only two
possible outcomes are possible (H, T). But when two coins
are tossed then there will be four possible outcomes, i.e {(H,
H), (H, T), (T, H), (T, T)}.

4. Applications of Probability
Probability has a wide variety of applications in real life.
Some of the common applications which we see in our
everyday life while checking the results of the following
events:
Choosing a card from the deck of cards
Flipping a coin
Throwing a dice in the air
Pulling a red ball out of a bucket of red and white balls
Winning a lucky draw

5. Other Major Applications of Probability
Other Major Applications of Probability
It is used for risk assessment and modelling in various
industries
Weather forecasting or prediction of weather changes
Probability of a team winning in a sport based on players
and strength of team
In the share market, chances of getting the hike of share
prices

6. Probability Terms and Definition

7. RULE OF ADDITION
› The addition rule for probabilities consists of two rules or
formulas, with one that accommodates two mutually-
exclusive events and another that accommodates two non-
mutually exclusive events.
› Non-mutually-exclusive means that some overlap exists
between the two events in question and the formula
compensates for this by subtracting the probability of the
overlap, P(Y and Z), from the sum of the probabilities of Y
and Z.
› In theory the first form of the rule is a special case of the
second form.

8. 1ST RULE: Mathematically, the probability of two
mutually exclusive events is denoted by:
P(Y or Z)=P(Y)+P(Z)
To illustrate the first rule in the addition rule for probabilities,
consider a die with six sides and the chances of rolling either
a 3 or a 6. Since the chances of rolling a 3 are 1 in 6 and the
chances of rolling a 6 are also 1 in 6, the chance of rolling
either a 3 or a 6 is:
1/6 + 1/6 = 2/6 = 1/3

9. 2ND RULE:Mathematically, the probability of two
non-mutually exclusive events is denoted by:
(Y or Z)=P(Y)+P(Z)−P(Y and Z)
To illustrate the second rule, consider a class in which there
are 9 boys and 11 girls. At the end of the term, 5 girls and 4
boys receive a grade of B. If a student is selected by chance,
what are the odds that the student will be either a girl or a B
student? Since the chances of selecting a girl are 11 in 20, the
chances of selecting a B student are 9 in 20 and the chances of
selecting a girl who is a B student are 5/20, the chances of
picking a girl or a B student are:
(Y or Z)=P(Y)+P(Z)−P(Y and Z)
11/20 + 9/20 - 5/20 =15/20 = 3/4

10. THE COMPLEMENT RULE
The Complement of an Event
› The complement A′ of the event A consists of all elements of
the sample space that are not in A.
Determining Complements of an Event
Let us refer back to the experiment of throwing one die. As you
know, the sample space of a fair die is S={1,2,3,4,5,6}. If we
define the event A as observing an odd number, then
A={1,3,5}. The complement of A will be all the elements of the
sample space that are not in A. Thus, A′={2,4,6}.

11. The Complement Rule states that the sum of
the probabilities of an event and its
complement must equal 1.
P(A)+P(A′)=1
› probability of the event, P(A), is calculated using the relationship
P(A)=1−P(A′)
1. Suppose you know that the probability of getting the flu this winter is 0.43.
What is the probability that you will not get the flu?
Let the event A be getting the flu this winter. We are given P(A)=0.43. The
event not getting the flu is A′. Thus,
P(A′)=1−P(A)
=1−0.43
P(A′) =0.57

12. Solved Examples
1. There are 6 pillows in a bed, 3 are red, 2 are yellow
and 1 is blue. What is the probability of picking a
yellow pillow?
Ans: The probability is equal to the number of yellow pillows
in the bed divided by the total number of pillows, i.e. 2/6 =
1/3.

13. CONDITIONAL RULE
Conditional probability is the probability of one event occurring with some
relationship to one or more other events.
Events in Conditional Probability
Conditional probability could describe an event like:
Event A is that it is raining outside, and it has a 0.3 (30%) chance of raining
today.
Event B is that you will need to go outside, and that has a probability of 0.5
(50%).
A conditional probability would look at these two events in relationship with
one another, such as the probability that it is both raining and you will need to
go outside.

14. CONDITIONAL RULE
› The formula for conditional probability is:
P(B|A) = P(A and B) / P(A)
› which you can also rewrite as:
P(B|A) = P(A∩B) / P(A)

15. › Example 1
› In a group of 100 sports car buyers, 40 bought alarm systems, 30
purchased bucket seats, and 20 purchased an alarm system and
bucket seats. If a car buyer chosen at random bought an alarm
system, what is the probability they also bought bucket seats?
› Step 1: Figure out P(A). It’s given in the question as 40%, or 0.4.
› Step 2: Figure out P(A∩B). This is the intersection of A and B: both
happening together. It’s given in the question 20 out of 100 buyers, or
0.2.
› Step 3: Insert your answers into the formula:
› P(B|A) = P(A∩B) / P(A) = 0.2 / 0.4 = 0.5
› The probability that a buyer bought bucket seats, given that they
purchased an alarm system, is 50%

16. MULTIPLICATION RULE
If Aand B are two independent events in a probability experiment, then the
probability that both events occur simultaneously is:
P(A and B)=P(A)⋅P(B)
In case of dependent events , the probability that both events occur
simultaneously is:
P(A and B)=P(A)⋅P(B | A)
(The notation P(B | A)means "the probability of B, given that A
has happened.")

17. Example 1:
› You have a cowboy hat, a top hat, and an Indonesian hat called a songkok.
You also have four shirts: white, black, green, and pink. If you choose one
hat and one shirt at random, what is the probability that you choose the
songkok and the black shirt?
› The two events are independent events; the choice of hat has no effect on
the choice of shirt.
› There are three different hats, so the probability of choosing the songkok is
1/3
› . There are four different shirts, so the probability of choosing the black
shirt is 1/4
› So, by the Multiplication Rule:
› P(songok and black shirt)=1/3 X1/4=1/12

18. EXAMPLE 2
› Suppose you take out two cards from a standard pack of
cards one after another, without replacing the first card.
What is probability that the first card is the ace of spades,
and the second card is a heart?The two events are
dependent events because the first card is not replaced.
› There is only one ace of spades in a deck of 52cards. So:
› P(1st card is the ace of spades)=1/52

19. Problems and Solutions on Probability
Question 1: Find the probability of ‘getting 3 on rolling a die’.
Solution:
Sample Space = S = {1, 2, 3, 4, 5, 6}
Total number of outcomes = n(S) = 6
Let A be the event of getting 3.
Number of favourable outcomes = n(A) = 1
i.e. A = {3}
Probability, P(A) = n(A)/n(S) = 1/6
Hence, P(getting 3 on rolling a die) = 1/6

20. Problems and Solutions on Probability
Question 2: Draw a random card from a pack of cards. What is the
probability that the card drawn is a face card?
Solution:
A standard deck has 52 cards.
Total number of outcomes = n(S) = 52
Let E be the event of drawing a face card.
Number of favourable events = n(E) = 4 x 3 = 12 (considered Jack, Queen and
King only)
Probability, P = Number of Favourable Outcomes/Total Number of Outcomes
P(E) = n(E)/n(S)
= 12/52
= 3/13
P(the card drawn is a face card) = 3/13

21. Problems and Solutions on Probability
Question 3: A vessel contains 4 blue balls, 5 red balls and 11 white
balls. If three balls are drawn from the vessel at random, what is the
probability that the first ball is red, the second ball is blue, and the
third ball is white?
Solution:
Given,
The probability to get the first ball is red or the first event is 5/20.
Since we have drawn a ball for the first event to occur, then the number of
possibilities left for the second event to occur is 20 – 1 = 19.
Hence, the probability of getting the second ball as blue or the second
event is 4/19.
Again with the first and second event occurring, the number of possibilities
left for the third event to occur is 19 – 1 = 18.
And the probability of the third ball is white or the third event is 11/18.
Therefore, the probability is 5/20 x 4/19 x 11/18 = 44/1368 = 0.032.
Or we can express it as: P = 3.2%.

22. Solution:
To find the probability that the
sum is equal to 1 we have to
first determine the
sample space S of two dice as
shown below.
S = {
(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(6,1),(6,2),(6,3),(6,4),(6,5),(6,6) }
So, n(S) = 36
Question 4: Two dice are rolled, find the
probability that the sum is:
equal to 1
equal to 4
less than 13
1) Let E be the event “sum equal to 1”. Since, there are no
outcomes which where a sum is equal to 1, hence,
P(E) = n(E) / n(S) = 0 / 36 = 0
2) Let A be the event of getting the sum of numbers on
dice equal to 4.
Three possible outcomes give a sum equal to 4 they are:
A = {(1,3),(2,2),(3,1)}
n(A) = 3
Hence, P(A) = n(A) / n(S) = 3 / 36 = 1 / 12
3) Let B be the event of getting the sum of numbers on
dice is less than 13.
From the sample space, we can see all possible outcomes for
the event B, which gives a sum less than B. Like:
(1,1) or (1,6) or (2,6) or (6,6).
So you can see the limit of an event to occur is when both
dies have number 6, i.e. (6,6).
Thus, n(B) = 36
Hence,
P(B) = n(B) / n(S) = 36 / 36 = 1

23. NORMAL PROBABILITY CURVE:
› The literal meaning of the term normal is average. Most of
the things like intelligence, wealth, beauty, height etc. are
quite equally distributed. There are quite a few persons
who deviate noticeably from average, either above or
below it. If we plot such a distribution on a graph paper, we
get a bell-shaped curve, referred to as Normal Curve.

24. Normal Curve/Gaussian Curve
› The data from a certain coin or a dice throwing experiment
involving a chance success or probability; if plotted on a graph
paper gives a frequency curve which closely resembles the normal
curve. Hence, it is also known as Normal Probability Curve.
› Normal curve was derived by Laplace and Gauss (1777-1855)
independently. They also named it ‘curve of error’, where ‘error’ is
used in the sense of a deviation from the normal, true value. In the
honour of Gauss, it’s also known as Gaussian Curve’.
› The normal curve takes into account the law which states that the
greater the deviation from the mean or an average, the less
frequently it occurs. For e.g. in terms of Intelligence, its rare to find
people with very low or very high intelligence. It’s normally
distributed in the population.

25. Normal Curve

27. › 1. 50% of the scores occur above the mean and
50% below.
› 2. Approximately 34% between the mean and 1
SD above mean.
› 3. Approximately 34% between the mean and 1
SD below mean.
› 4. Approximately 68% of all scores occur
between the mean & +/-1SD
› 5. Approximately 95% between the mean and
+/-2 SDs.
› 6. Approximately 99% of the scores fall
between -3 and +3 SDs.
› 7. The area on the normal curve between 2 and
3 SDs above & 2 and 3 SDs below the mean are
› known as tails.
› 8. The normal curve has 2 tails.

28. CHARACETRISTICS OF NORMAL CURVE:
›a) For this curve, mean, median and mode are the same.
›b) The curve is perfectly symmetrical. In the sense, it is not skewed. The value of measured
›skewness for normal curve is zero.
›c) The normal curve serves as a model for describing the flatness or peakedness of a curve
through the measure of kurtosis. For the normal curve, the value of kurtosis is 0.263. April 9, 2020
›d) The curve is asymptotic. It approaches but never touches the X-axis. It is because of the
›possibility of locating in the population a case which scores still higher than the highest score or
›still lower than the lowest score. Therefore, theoretically, it extends from minus infinity to plus
›infinity.
›e) As the curve does not touch the base line, the mean is used as the starting point for working
›with the normal curve.
›f) To find out deviations from the mean, standard deviation of the distribution (σ) is used as a unit
›of measurement.
›g) The curve extends on both sides -3σ distance on the left to +3σ on the right