Probability definition

It is a branch of mathematics that deals with calculating the chance of a given event’s occurrence. It can be expressed as a number between 1 and 0.

Probability theorems….

· Rule of Subtraction

P(A) = 1 – P(A’)

Example

The probability that Tom will graduate from school is 0.50. What is the probability that Tom will not graduate from school? Based on the rule of subtraction, the probability that Tom will not graduate is 1.00 – 0.50 or 0.50.

· Rule of Multiplication

P(A ? B) = P(A) P(B|A)

Example

A box contains 5 green cards and 3 pink cards. Two cards are drawn without replacement from the urn. What is the probability that both of the cards are pink?

Solution: Let A = the event that the first card is pink; and let B = the event that the second card is pink. We know the following:

In the beginning, there are 8 cards in the box, 3 of which are pink. Therefore, P(A) = 3/8.

After the first selection, there are 7 cards in the box, 2 of which are pink. Therefore, P(B/A) = 2/7.

Therefore, based on the rule of multiplication:

P(A ? B) = P(A) P(B|A)

P(A ? B) = (3/8) * (2/7) = 6/56 = 3/28

· Rule of Addition

P(A ? B) = P(A) + P(B) – P(A ? B)

Example

A girl goes to the shop. The probability that she buys (a) a book is 0.40, (b) a notebook is 0.30, and (c) both book and notebook is 0.20. What is the probability that the student buys a book, a notebook, or both?

Answer

Let F = the event that the student bus a book; and let N = the event that the student buys a notebook.

By applying the rule of addition

P(F ? N) = P(F) + P(N) – P(F ? N)

P(F ? N) = 0.40 + 0.30 – 0.20 = 0.50

Types of random variables

· A discrete random variable is one which may take on only a countable number of distinct values, for example: (0,1,2,3,4,…….)

Discrete random variables are usually counted. If a random variable can take only a partial number of distinct values, then it must be discrete. Examples of discrete random variables include the number of children in a school, the Sunday night attendance at a cinema, and the number of patients in a hospital.

· A continuous random variable is one which takes an endless number of possible values, and it is usually measurements. Examples include height, the amount of sugar in a juice, the time required to run a kilo.

A continuous random variable is not defined by limited values. Instead, it is defined over an interval of values. It is represented by the area under a curve (in higher mathematics, this is known as an integral). The probability of observing any single value is 0 “since the number of values which may be assumed by the random variable is infinite”.

Probability Distribution

A probability distribution tells you what the chance of an event happening is. Probability distributions can show simple events. For example, tossing a coin or picking a card. Not only simple events but also it can show much more difficult events, like the probability of a certain drug successfully treating cancer.

Different types of probability distribution…

· Discrete Probability Distribution

A discrete probability distribution consists of discrete variables. A random variable will get a discrete probability distribution if it is discrete.

1. Binomial distributions. They have “Successes” and “Failures.”

2. A Poisson distribution is a mean that helps to expect the probability of certain events. It gives us the chance of a given number of events happening in a fixed period of time.

3. Hypergeometric distribution which describes the times of successes in the first m of a series of n consecutive Yes/No experiments, and this distribution arises when there is no replacement.

· Continuous probability distribution

The continuous probability distribution is a type of distributions that deals with continuous types of data or random variables.

1. Normal distributions which describe those tests that are basic methods of finding the possible differences between two samples.

2. The Exponential distribution is a one-parameter distribution that is a case of the gamma distribution. It is usually used in survival studies. Its density function is given below

f(t)= exp(?t/?)/?

Note that both ? and t>0.

3. The Cauchy distribution is describing resonance behavior, and it also describes the distribution of horizontal distances at which a line segment inclined at a random angle cuts the x-axis