Probability
What is Probability?
Probability is a way of measuring how likely an event is to happen. It is found by dividing the number of outcomes you want (favourable outcomes) by the total number of possible outcomes.
Formula:
Probability of an event = Favourable outcomes ÷ Total outcomes = x / n
For example, we use probability to predict the result of tossing a coin, rolling a die, or picking a card from a deck.
There are two main types of probability:
- Theoretical probability — based on reasoning and known facts.
Experimental probability — based on actual experiments or observations.
Events in Probability
In probability, an event is simply a group of possible results from an experiment. These results come from something called the sample space, which is the set of all possible outcomes.
If P(E) shows the probability of an event E, then:
- P(E) = 0 → The event can never happen (impossible event).
- P(E) = 1 → The event will definitely happen (certain event).
- 0 ≤ P(E) ≤ 1 → Probability is always between 0 and 1.
If we have two events A and B, then:
- P(A) > P(B) means A is more likely to happen than B.
Here’s the formula:
- P(E) = n(E) / n(S) → Number of outcomes for event E ÷ Total outcomes in the sample space.
For the event not happening (called E’):
- P(E’) = [n(S) − n(E)] / n(S) = 1 − P(E)
This means:
- P(E) + P(E’) = 1 → The probability of something happening and not happening always adds up to 1.
Types of Probability
There are three important classes of probabilities:
- Theoretical Probability
- Experimental Probability
- Axiomatic Probability
Let us learn about three of them in the section:
Theoretical Probability
Theoretical Probability is based on the potential chances of something happening. The type of probability is principally based on the logic behind probability. For example, if a coin is tossed, the theoretical probability of getting a head or a tail will be ½ or o.5.
Experimental Probability
This type of probability is based on the observations of an experiment. The experimental probability can be determined based on the number of potential outcomes by the cumulative/total number of trials. For instance, if a coin is tossed 8 times and the head is recorded 4 times then, the experimental probability for heads is 4/8 or 1/2.
Axiomatic Probability
In axiomatic probability, a set of commands or assumptions are set which fits all types. These axioms are set by Kolmogorov and are called Kolmogorov’s three axioms. With the axiomatic method of probability, the chances of existence or non-existence of the events can be quantified.
Also, read about Relations and Functions here.
Complementary Events
The possibility that there will be simply two outcomes that state that an event will occur or will not occur. For example, a person will visit or not visit your residence, getting a government job or not receiving a job, etc. are examples of complementary events. The complement of an event occurring is exactly the reverse of the probability of it not occurring. Some more additional examples are:
- It will rain or not rain tomorrow.
- The trainee will pass the examination or not pass.
- You win the quiz or you won’t win.
Learn about Multiplication Theorem of Probability
Probability Density Function
The Probability Density Function (PDF) is the probability function that is outlined for the density of a continuous random variable existing within a certain range of values. The PDF explains the normal distribution and how mean and deviation exist. The standard normal distribution is applied to build a database of statistics, which are frequently used in science to describe the real-valued variables, whose distribution is not identified.
Learn about Sequences and Series and Rolling a Die
Probability Formulas
In probability examples, one thing that helps a lot are the formulas and theorem as probability sometimes gets a little confusing, so next will look at the formulas;
- P(A ∪ B) = P(A) + P(B) — P(A ∩ B).
- If A and B are mutually exclusive events i.e A ∩ B = ϕ, then P(A ∪ B) = P(A) + P(B).
- If A, B, and C are any three events then the formula for addition is given by;
- P(A ∪ B ∪ C) = P(A) + P(B) + P(C) — P(A ∩ B) — P(B ∩ C) — P(A ∩ C) + P(A ∩ B ∩ C)
- If A, B, and C are any three events such that they are mutually exclusive then;
- P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
Proceeding further is the concept of conditional probability.
Learn about Difference Between Mutually Exclusive and Independent Events
Conditional Probability
Let A and B be any two events correlated with a random experiment. Then, the probability of occurrence of an event A with the condition that B has already occurred such that the probability of B is not equal to zero (P(B) ≠ 0), is called the conditional probability and denoted by P(A|B).
P(A|B) = P(A ∩ B) / P(B)
Similarly, P(B|A) = P(A ∩ B) / P(A), where P(A) ≠ 0
Check more topics of Mathematics here.
Key points for Conditional Probability
Consider if E̅ is the complement of the event E, then
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P(E̅) = 1 − P(E)
For any event A, 0 ≤ P(A) ≤ 1, this is the range of probability.
If A and B are independent events, then P(B|A) = P(B) ∵ P(A ∩ B) = P(A) × P(B)
Points to remember for Probability
If A and B are any two events in a random experiment then;
P(A ∩ B) = P(A | B) P(B), if P(B) ≠ 0.
P(A ∩ B) = P(B | A) P(A), if P(A) ≠ 0.
Coin Toss Probability
When we toss a coin, there are two possible results — Head (H) or Tail (T). In cricket and other games, coin tosses are often used to make quick decisions, like who will bat or bowl first.
Tossing One Coin
- Possible outcomes: 2 → {H, T}
- Probability of Head: P(H) = 1/2
- Probability of Tail: P(T) = 1/2
Tossing Two Coins
- Possible outcomes: 4 → {(H,H), (H,T), (T,H), (T,T)}
- Probability of 2 Heads: P(2H) = 1/4
- Probability of 1 Head: P(1H) = 2/4 = 1/2
- Probability of 0 Heads: P(0H) = 1/4
Tossing Three Coins
- Possible outcomes: 8 → {(H,H,H), (H,H,T), (H,T,H), (T,H,H), (T,T,H), (T,H,T), (H,T,T), (T,T,T)}
- P(0 Heads) = 1/8
- P(1 Head) = 3/8
- P(2 Heads) = 3/8
- P(3 Heads) = 1/8
Dice Roll Probability
Dice are used in many games. A single die has 6 faces numbered 1 to 6.
Rolling One Die
- Sample space: {1, 2, 3, 4, 5, 6}
- P(Even number) = 3/6 = 1/2
- P(Odd number) = 3/6 = 1/2
- P(Prime number: 2, 3, 5) = 3/6 = 1/2
Rolling Two Dice
- Total outcomes = 6 × 6 = 36
- P(Doublet: same number) = 6/36 = 1/6
- P(At least one die shows 3) = 11/36
- P(Sum is 7) = 6/36 = 1/6
Formula for outcomes when rolling n dice = 6ⁿ
Probability of Drawing Cards
A deck has 52 cards in 4 suits: Clubs (♣), Diamonds (♦), Hearts (♥), Spades (♠).
- Each suit has 13 cards: Ace, 2–10, Jack, Queen, King.
- Black cards: Spades, Clubs (26 cards)
- Red cards: Hearts, Diamonds (26 cards)
- Face cards: Jack, Queen, King (12 cards)
Examples:
- P(Black card) = 26/52 = 1/2
- P(Hearts) = 13/52 = 1/4
- P(Face card) = 12/52 = 3/13
- P(Card with number 4) = 4/52 = 1/13
- P(Red card with number 4) = 2/52 = 1/26
Probability Tree Diagram
A tree diagram is a simple picture that shows all the possible outcomes of an event and helps us find their probabilities. It’s called a “tree” because it starts from one point and then branches out, just like the branches of a tree.
For example, when we toss a coin, there are two possible outcomes — Heads (H) or Tails (T). In a tree diagram, each branch represents one outcome, and we write its probability on that branch (like 0.5 for both heads and tails).
An important rule to remember is:
The probabilities of all branches coming from the same point will always add up to 1.
For a coin toss: 0.5 (Heads) + 0.5 (Tails) = 1.
Probability Theorems
These theorems make it easier to understand how probability works and to solve problems related to it.
- Event and its Complement — The probability of an event happening plus the probability of it not happening is always 1.
Formula: P(A) + P(A′) = 1 - Impossible Event — The probability of something that cannot happen is always 0.
Formula: P(ϕ) = 0 - Sure Event — If something is certain to happen, its probability is 1.
Formula: P(A) = 1 - Probability Range — The probability of any event will always be between 0 and 1.
Formula: 0 ≤ P(A) ≤ 1 - Two Events (A and B) –
- For any two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- If the events are mutually exclusive (cannot happen together): P(A ∪ B) = P(A) + P(B)
