Probability Basics : Lesson 1

Today we explore the basic concepts of Probability.

Probability is the measure of the likelihood of an "event" to occur.
For example, when we toss a fair coin, the chances of a head turning up are 1/2. When we roll a dice the chance of 2 appearing is 1/6.

Important Terms:

1. Sample Space : The collection of all results is called the sample space of the event.
For example, when a coin is tossed, the Sample space(S) is the set {Head, Tails}.
When a dice is rolled, S = {1,2,3,4,5,6}

2. Event: A subset of the S is called an event. For example, when a coin is tossed, {Head}, {Tail} are events. Note that an event set can contain multiple items. For example when 2 coins are tossed, {Head,Head} is considered an event.

3. Equally Probable Events: When 2 events have the same likelihood of occuring, they are known as Equally Probable events. For example the chance of head and tails turning up on the toss of a coin are Equally Probable events.

Probability

The Probability of an event occurring is defined as the number of cases favorable for the event divided by total number of events in the sample space.

If the event be called as 'A', the probability is represented as P(A)

Example: Probability that head shows up on tossing a coin

Favorable Cases : Head (1)

Sample Space: Head, Tail (2)

P(A) = Favorable Cases/Sample Space = 1/2

P(A') is used to represent the probability of event A not occurring.

Note: P(A) + P(A') = 1

If we have 2 events A,B as the overall sample space, then:

P(A) = A / (A+B) and P(B) = B / (A + B)

Independent Events and Mutually Exclusive Events

A regular confusion among students is the difference between Independent and Mutually exclusive Events. 

Let A, B be 2 events. P(A and B) = P (A ∩ B ) is defined as the probability that both A and B occur together.

For 2 independent events A, B, P(A ∩ B ) = P(A) * P(B)

For Mutually Exclusive events, P(A ∩ B ) = 0

This is best understood with an example.

Consider a fair coin and a fair six-sided die. Let event A be obtaining tails, and event B be rolling a 3. Then we can safely say that events A and B are independent, because the outcome of one does not affect the outcome of the other.

Here, P(A ∩ B ) = 1/2 * 1/6 = 1/12.
Here A and B are Independent Events (Not mutually exclusive).


Consider a fair six-sided dice, where even-numbered faces are colored red, and the odd-numbered faces are colored green. Let event A be rolling a green face, and event B be rolling a 6.

P(A) = 3/6 = 1/2
P(B) = 1/6

But note that A&B cannot occur simultaneously since 6 is always going to turn up on a red face.

Here, P(A ∩ B ) =0
Here A and B are Mutually exclusive events(Not independent).


In our next post we will go deeper into complex probability theory and solve a few problems and provide video solutions for the same.