Probability and Probability Distributions

Introduction

Probability is the chance that an event will occur.
For equally likely events, it is the number of ways an event can occur divided by number of outcomes in the sample space.


Event is a subset of the sample space having one or more outcomes.

Probability and Probability Distributions

Equally likely events include events which have the same chance of occurring.
On the other hand, complementary events are two mutually exclusive events that are all inclusive.

Sample Spaces

The sample space is the set of all the possible outcomes in an experiment and is denoted by a capital letter S.
For example, if a single die is rolled, then S = { 1, 2, 3, 4, 5, 6 }, which represents the set of all possible outcomes.
In case of two dice, the sum of the two dice, then S = { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }.

Independent and Dependent Events

Independent events are events that do not affect each other and the occurrence of one event does not change the probability of the occurrence of the other.
Dependent Events are events that are not independent and depend on each other.

Mutually Exclusive and Inclusive Events

Mutually exclusive events are disjoint events that can not happen at the same time.
All inclusive events are events whose union includes the totality of the sample space.

Probability of a single event

If event A can occur in nA ways out of a total of N possible and equally likely outcomes,
the probability that event A will occur is given by the following formula:
P(A) = nA / N

where: If P(A) = 0, event A cannot occur. If P(A) = 1, event A will occur with certainty.
If P(A') represents the probability of nonoccurrence of event A, P(A)+P(A')=1

Examples

Example 1

A head (H) and a tail (T) are the two equally possible outcomes in tossing a fair (balanced) coin.
Hence, P(H) = nH / N = 1/2
P(T) = nT / N = 1/2
P(H) + P(T) = 1

Example 2

When rolling a balanced die once, there are six possible and equally likely outcomes: 1, 2, 3, 4, 5, and 6.
All probabilities are equal: P(1)= P(2) = P(3) = P(4) = P(5) = P(6) = 1/6
Question: What is the probability of not rolling a 1?
Solution:
P(1) + p(1') = 1/6 + p(1') = 6/6; which implies that p(1') = 5/6

Example 3

Let's assume that in 100 tosses of a fair coin, we get 63 heads and 37 tails.
The relative frequency of tails is 37/100, or 0.37, which represents the relative frequency or empirical probability.
This relatice frequency is totally different from the classical probability of P(T) = 0.5.

Example 4

If we keep increasing the number of tosses and approaches infinity in the limit, in this case,
the relative frequency or empirical probability gets close to the classical probability.

For example, the relative frequency or empirical probability might be 0.52 for 500 tosses, 0.51 for 1,000 tosses, and 0.505 for 10,000 tosses, and so on.


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Date of last modification: March 25, 2019