# NORMAL DISTRIBUTION

PROBABILITY

Probability:

Probability is used to measure the ‘likelihood’ or ‘chances’ of certain events (pre-specified outcomes) of an experiment.

If an event can occur in N mutually exclusive and equally likely ways, and if m of these possess a trait E, the probability of the occurrence of E expressed as:

Number of favourable cases

=

Total number of outcome (sample Space)

Characteristics of probability:

• It is usually expressed by the symbol ‘P’
• It ranges from 0 to 1
• When P = 0, it means there is no chance of happening or impossible.
• If P = 1, it means the chances of an event happening is 100%.
• The total sum of probabilities of all the possible outcomes in a sample space is always equal to one (1).
• If the probability of occurrence is p(o)= A, then the probability of non-occurrence is 1-A.

Terminology

Random Experiment:

Any natural phenomenon, yielding some result will be termed as random experiment when it is not possible to predict a particular result to turn out.

An Outcome:

The result of an experiment in all possible form are said to be event of that experiment. e.g. When you toss a coin once, you either get head or tail.

A trial:

This refers to an activity of carrying out an experiment like tossing a coin or rolling a die or dices.

Sample Space:

A set of All possible outcomes of a probability experiment.

Example 1: In tossing a coin, the outcomes are either Head (H) or tail (T) i.e. there are only two possible outcomes in tossing a coin. The chances of obtaining a head or a tail are equal. It can be solved as follow;

n(s) = 2 ways

S = {H, T}

Example 2: what is sample space when single dice is rolled?

n(s) = 6 ways

S = {1, 2, 3, 4, 5, 6}

A Simple Event

In an experimental probability, an event with only one outcome is called a simple event.

Compound Events

When two or more events occur in connection with each other, then their simultaneous occurrence is called a compound event.

Mutually exhaustive:

If in an experiment the occurrence of one event prevents or rules out the happening of all other events in the same experiment then these event are said to be mutually exhaustive events.

Mutually exclusive:

Two events are said to be mutually exclusive if they cannot occur simultaneously.

Example: tossing a coin, the events head and tail are mutually exclusive because if the outcome is head then the possibilities of getting a tail in the same trial is ruled out.

Equally likely events:

Events are said to be equally likely if there is no reason to expect any one in preference to other.

Example: in a single cast of a fair die each of the events 1, 2, 3, 4, 5, 6 is equally likely to occur.

Favourable case:

The cases which ensure the occurrence of an event are said to be favourable to the events.

Independent event:

When the experiments are conducted in such a way that the occurrence of an event in one trial does not have any effect on the occurrence of the other events at a subsequent experiment, then the events are said to be independent.

Example:

If we draw a card from a pack of cards and again draw a second a card from the pack by replacing the first card drawn, the second draw is known as independent f the first.

Dependent Event:

When the experiments are conducted in such a way that the occurrence of an event in one trial does have some effect on the occurrence of the other events at a subsequent experiment, then the event are said to be dependent event.

Example:

If we draw a card from a pack and again draw a card from the rest of pack of cards (containing 51 cards) then the second draw is dependent on the first.

Conditional Probability:

The probability of happening of an event A, when it is known that B has already happened, is called conditional probability of A and is denoted by P (A/B) i.e.

• P(A/B) = conditional probability of A given that B has already occurred.
• P (A/B) = conditional Probability of B given that A has already occurred.

Types of Probability:

The Classical or mathematical:

Probability is the ratio of the number of favorable cases as compared to the total likely cases.

The probability of non-occurrence of the same event is given by {1-p (occurrence)}.

The probability of occurrence plus non-occurrence is equal to one.

If probability occurrence; p (O) and probability of non-occurrence (O’), then p(O)+p(O’)=1.

Statistical or Empirical

Empirical probability arises when frequency distributions are used. For example:

 Observation ( X) 0 1 2 3 4 Frequency ( f) 3 7 10 16 11

The probability of observation (X) occurring 2 times is given by the formulae

RULES OF PROBABILITY

1. Rule 1: When two events A and B are mutually exclusive, then probability of any one of them is equal to the sum of the probabilities of the happening of the separate events;

Mathematically:

P (A or B) =P (A) +P (B)

Example: When a die or dice is rolled, find the probability of getting a 3 or 5.

Solution: P (3) =1/6 and P (5) =1/6.

Therefore P (3 or 5) = P (3) + P (5) = 1/6+1/6 =2/6=1/3.

2) Rule 2: If A and B are two events that are NOT mutually exclusive, then

P (A or B) = P(A) + P(B) – P(A and B), where A and B means the number of outcomes that event A and B have in common.

Given two events A and B, the probability that event A, or event B, or both occur is equal to the probability that event A occurs, plus the probability that event B occurs, minus the probability that the events occur simultaneously.

Example: When a card is drawn from a pack of 52 cards, find the probability that the card is a 10 or a heart.

Solution: P (10) = 4/52 and P (heart) =13/52

P (10 that is Heart) = 1/52

P (A or B) = P (A) +P (B)-P (A and B) = 4/52 _ 13/52 – 1/52 = 16/52.

Multiplication Rule

1. Rule 1: For two independent events A and B, then

P (A and B) = P (A) x P (B).

Example: Determine the probability of obtaining a 5 on a die and a tail on a coin in one throw.

Solution: P (5) =1/6 and P (T) =1/2.

P (5 and T) = P (5) x P (T) = 1/6 x ½= 1/12.

1. Rule 2: When two events are dependent, the probability of both events occurring is P (A and B) =P (A) x P (B|A), where P (B|A) is the probability that event B occurs given that event A has already occurred.

Example: Find the probability of obtaining two Aces from a pack of 52 cards without replacement.

Solution: P (Ace) =2/52 and P (second Ace if NO replacement) = 3/51

Therefore P (Ace and Ace) = P (Ace) x P (Second Ace) = 4/52 x 3/51 = 1/221

Construct sample space, when two dice are rolled

n(s) = n1 x n2 = 6 x 6 = 36

 (1,1) (2,1) (3,1) (4,1) (5,1) (6,1) (1,2) (2, 2) (3, 2) (4, 2) (5, 2) (6, 2) (1, 3) (2, 3) (3, 3) (4, 3) (5, 3) (6, 3) (1, 4) (2, 4) (3, 4) (4, 4) (5, 4) (6, 4) (1, 5) (2, 5) (3, 5) (4, 5) (5, 5) (6, 5) (1, 6) (2, 6) (3, 6) (4, 6) (5, 6) (6, 6)

EXAMPLE OF FINDING PROBABILITY OF AN EVENT

If 3 coins are tossed together, construct a tree diagram & find the followings;

n (s) = n1 x n2 x n3

= 2 x 2 x2 = 8

1. Event showing no head = P(X = 0)

1. Event showing 01 head = P(X = 1)

Answer: HTT, THT, TTH 3/8 = 0.375

1. Event showing 02 heads = P(X = 2)

Answer: HHT, HTH, THH 3/8 = 0.375

1. Event showing 03 heads = P(X = 3)

Complementary Events

Complementary events happen when there are only two outcomes, like getting a job, or not getting a job. In other words, the complement of an event happening is the exact opposite: the probability of it not happening.

The probability of not occurrence of an event.

The probability of an event A is equal to 1 minus the probability of its complement, which is written as Ā and

P (Ā) = 1 – P (A)

CONDITIONAL PROBABILITY &SCREENING TESTS

Sensitivity, Specificity, and Predictive Value Positive and Negative

In the health sciences field a widely used application of probability laws and concepts is found in the evaluation of screening tests and diagnostic criteria. Of interest to clinicians is an enhanced ability to correctly predict the presence or absence of a particular disease from knowledge of test results (positive or negative) and/or the status of presenting symptoms (present or absent). Also of interest is information regarding the likelihood of positive and negative test results and the likelihood of the presence or absence of a particular symptom in patients with and without a particular disease.

In consideration of screening tests, one must be aware of the fact that they are not always infallible. That is, a testing procedure may yield a false positive or a false negative.

False Positive:

A false positive results when a test indicates a positive status when the true status is negative.

False Negative:

A false negative results when a test indicates a negative status when the true status is positive.

Sensitivity:

The sensitivity of a test (or symptom) is the probability of a positive test result (or presence of the symptom) given the presence of the disease.

Specificity:

The specificity of a test (or symptom) is the probability of a negative test result (or absence of the symptom) given the absence of the disease.

Predictive value positive:

The predictive value positive of a screening test (or symptom) is the probability that a subject has the disease given that the subject has a positive screening test result (or has the symptom).

Predictive value negative:

The predictive value negative of a screening test (or symptom) is the probability that a subject does not have the disease, given that the subject has a negative screening test result (or does not have the symptom).

Summary of formulae:

Symbols

COUNTING RULES

1) FACTORIALS (number of ways)

The result of multiplying a sequence of descending natural numbers down to a number. It is denoted by “!”

Examples:

4! = 4 × 3 × 2 × 1×0! = 24

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

Remember : 0! = 1

General Method:

n! = n (n -1) (n -2) (n -3)……….. (n – n)!

2) PERMUTATION RULES

All possible arrangements of a collection of things, where the order is important in a subset.

Repetition of same items with different arrangement is allowed.

Examples

1. COMBINATIONS

The order of the objects in a subset is immaterial.

Repetition of same objects in not allowed with different arrangement

Examples:

Binomial distribution:

Binomial distribution is a probability distribution which is obtained when the probability ‘P’ of the happening of an event is same in all the trials and there are only two event in each trial.

Conditions:

• Each trial results in one of two possible, mutually exclusive, outcomes. One of the possible outcomes is denoted (arbitrarily) as a success, and the other is denoted a failure.
• The probability of a success, denoted by p, remains constant from trial to trial. The probability of a failure (1 – p) is denoted by q.
• The trials are independent; that is, the outcome of any particular trial is not affected by the outcome of any other trial.
• Parameter should be available; (n & p) are parameters.

Formula:

b (X: n, p) = nCx px qn – x (OR) f (x) = nCx px qn – x

Where

X = Random variable

n = Number of Trials

p = Probability of Success

q = Probability of Failure

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