Introduction to Probability

Ever heard about a weather forecast at the end of a news bulletin on TV or read about the weather conditions of your city/country for the next few days in any newspaper? They specifically use the term “probability.” We are going to learn a few basic concepts, probability formulas involved to calculate the probability for different types of situations.

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Definition of Probability

Probability is the measure of uncertainty of any event (any phenomenon happened or bound to happen). Before we dive into the world of understanding the concept of Probability through the various formulas involved to calculate it, we need to understand few crucial terms or make ourselves familiar with the terminology associated with the Probability.

Browse more Topics Under Probability

• Introduction to Probability
• Probability of an Event
• Events and its Types
• Events and Its Algebra
• Independent Events
• Conditional Probability
• Basic Theorems of Probability
• Multiplication Theorem on Probability
• Baye’s Theorem
• Random Variable and Its Probability Distribution
• Mean and Variance of Random Distribution
• Bernoulli Trials and Binomial Distribution

• Experiment: Any phenomenon like rolling a dice, tossing a coin, drawing a card from a well-shuffled deck, etc.
• Outcome: The Result of any event; like number appearing on a dice, side of a coin, drawn out card, etc.
• Sample Space: The set of all possible outcomes.
• Event: Any combination of possible outcomes or the subset of sample space; like getting an even number on rolled dice, getting a head/tail on a flipped coin, drawing out a king/queen/ace of any suit.
• Probability Function: A function giving the probability for each outcome.

(Source: Wikipedia)

Probability Formulas

Probability = (Number of a Favourable outcome) / (Total number of outcomes)

                    P = n (E) / n (S)

Where P is the probability, E is the event and S is the sample space. Now, let’s looks at some very common examples.

Example 1: Probability of getting an even number on rolling a dice once.
Solution: Sample Space (S) = {1, 2, 3, 4, 5, 6}
Event (E) = {2, 4, 6}
Therefore, n (S) = 6 and n (E) = 3
Putting this in the probability formula, we get:
P = 3 / 6 = 1 / 2 = 0.5
This means, that the chances of getting an even number upon rolling a dice is 0.5

Example 2: Probability of getting HEAD at least once on tossing a coin twice.
Solution: Sample Space (S) = {HH, HT, TH, TT}; where H denotes Head and T denotes Tail.
Event (E) = {HH, HT, TH}
Therefore, Therefore, n (S) = 4 and n (E) = 3
Putting this in the probability formula, we get:
P = 3 / 4 = 0.75
This means, that the chances of getting at least one HEAD on tossing a coin twice are 0.75

Odds in Favour of the Event

Odds in the favor of any event is the ratio of the number of ways that an outcome can occur to the number of ways it cannot occur. Let’s look at an example.

Example 3: If a represents the odds in favor of getting number 4 on a single roll of dice & b represents the outcomes of not getting 4, then,

n (a) = Number of favorable outcomes = 1
n (b) = Number of favorable outcomes = (6 – 1) = 5
Odds in favor = 1 : 5 or 1 / 5
Probability (P) = Number of favorable outcomes/(Number of favorable outcomes + Number of unfavorable outcomes)
P = 1 / (1 + 5) = 1 / 6

Odds Against the Event

Odds against any event is the ratio of the number of ways that an outcome cannot occur to the number of ways it can occur. Let’s understand it through an example.

Example 4:  If a represents the odds against getting number 4 on a single roll of dice & b represents the outcomes of getting 4, then –

n (a) = Number of favorable outcomes = 1
n (b) = Number of favorable outcomes = (6 – 1) = 5
Odds in favor = 5 : 1 or 5 / 1
Probability (P) = Number of favorable outcomes / (Number of favorable outcomes + Number of unfavorable outcomes)
P = 5 / (1 + 5) = 5 / 6

Important Probability Formulas

I. Event (A OR B)

Also given by P (A U B) = P (A) + P (B) – P (A ∩ B)

If A & B are two mutually exclusive events then P (A ∩ B) = 0 and P (A U B) = P (A) + P (B). For example,

A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}
B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}
Thus, (A U B) = P (A) + P (B) = {1, 2, 3, 4, 5, 6}

II. Event (A AND B)

Also given by P (A ∩ B) = P (A) . P (B)

It gives the common elements that form the individual subsets of events A and B. For example,

A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}
B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}
Thus, (A ∩ B) = P (A) . P (B) = {4}

III. Event (A but NOT B)

For example,
A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}
B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}
Thus, (A but NOT B) = A – B = {1, 2, 3}; elements common in A and B get eliminated from A.

IV. Event (B but NOT A)

For example,
A = {Numbers greater than or equal to 4 in a dice roll} = {4, 5, 6}
B = {Numbers lesser than or equal to 4 in a dice roll} = {1, 2, 3, 4}
Thus, (B but NOT A) = B – A = {5, 6}; elements common in A and B get eliminated from B.

V. Event (NOT A)

The probability of occurrence of an event – P(A) then the probability of non-occurrence of the same event is P(A’). Some probability formulas based on them are as follows:

• P(A.A’) = 0
• P(A.B) + P (A’.B’) = 1
• P(A’B) = P(B) – P(A.B)
• P(A.B’) = P(A) – P(A.B)
• P(A+B) = P(AB’) + P(A’B) + P(A.B)

VI. Conditional Probability

P (B/A): Probability (conditional) of event B when event A has occurred.
P (A/B): Probability (conditional) of event A when event B has occurred.
P (A ∩ B) = P (A) . P (B/A)

These are some of the formulas that will help you solve mathematical problems on Probability.

Solved examples for You

Question: Find the probability of getting an even number greater than or equal to 4 in a dice roll.

Solution: Sample space (S) = {1, 2, 3, 4, 5, 6} and E = {4, 6}
P (E) = n (E) / n (S)
= 2 / 6
P (E) = 1 / 3

Question: Find the probability of getting at least one HEAD in a double coin toss.

Solution: S = {HH, HT, TH, TT}
E = {HH, TH, HT}
P (E) = n (E) / n(S)
So, P (E) = 3 / 4

This concludes our discussion on the topic of probability formulas.

Question: What is a simple probability?

Answer: It refers to the ratio of a number of outcomes favourable for an event to the total number of possible outcomes. In simple words, it is the measure of the possibility of an event or measure of the chance of an event to happen.

Question: What do U and ∩ mean in probability?

Answer: U simply means uniform distribution in probability. On the other hand, ∩ refers to the intersection of sets: Besides, the intersection of two given sets is the largest set that covers all the elements that are common to both the sets.

Question: State the four rules of probability?

Answer: Four rules of probability are:

1. Happening or non-happening: the possibility of happening and non-happening is always 1.
2. Exclusivity: Means that if two events A and B happen at the same time then they are mutually exclusive
3. Independence: Means that when one happens the other will not happen and visa-versa.
4. Sub-events: When one happens the other also happen.

Question: State the types of probability distributions?

Answer: Some of the types of probability distributions are:

1. Normal distribution
2. Chi-Square distribution
3. Binomial distribution
4. Poisson distribution

However, each probability distribution serves a different purpose and represent different data generation process.