Probability Calculator

Work out probabilities the NCERT Class 9 and 10 way: P(E) = favourable outcomes ÷ total outcomes, with the complement P(not E) = 1 − P(E). Pick a ready-made experiment — tossing coins, throwing one or two dice, drawing a card from a deck of 52 — to see the sample space and the favourable outcomes listed, or combine two events with the addition and multiplication rules. Every answer is shown as a fraction, a decimal and a percentage.

Experiment

Event

P(sum is 7)

1/6= 0.1667 = 16.67%

P(not E) = 1 − P(E)

5/6= 0.8333 = 83.33%

Favourable / total

6 / 36

Odds in favour

6 : 30

Event type

Less likely to happen than not

Step-by-step solution

Sample space for “throw two dice”: 36 outcomes (1,1), (1,2), …, (6,6) → 36 equally likely outcomes
Favourable outcomes for “sum is 7”: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6
P(E) = number of favourable outcomes ÷ total number of outcomes = 6/36 = 1/6
As a decimal: 0.1667; as a percentage: 16.67%
P(not E) = 1 − P(E) = 1 − 1/6 = 5/6 (complement)
Odds in favour = favourable : unfavourable = 6 : 30
Less likely to happen than not

How to use the probability calculator

Choose “Coins, dice & cards”, “One event” or “Two events”.
Coins, dice & cards: pick the experiment and the event, e.g. “Throw two dice” and “Sum is 7”.
One event: type the number of favourable outcomes and the total number of equally likely outcomes.
Two events: type P(A) and P(B) (1/2, 0.5 or 50%) and say whether they are independent, mutually exclusive or give P(A and B).

Formulas

Probability of an event	`P(E) = favourable outcomes ÷ total outcomes` 0 ≤ P(E) ≤ 1; impossible event 0, sure event 1
Complement	`P(not E) = 1 − P(E)`
Addition rule	`P(A or B) = P(A) + P(B) − P(A and B)`
Mutually exclusive	`P(A and B) = 0, P(A or B) = P(A) + P(B)`
Independent events	`P(A and B) = P(A) × P(B)`
Conditional probability	`P(A \| B) = P(A and B) ÷ P(B)`

Solved examples

Example 1: Two dice are thrown. Find the probability that the sum is 7.

Total outcomes = 6 × 6 = 36
Favourable: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6

Answer: 6/36 = 1/6 ≈ 16.67%

Example 2: A card is drawn from a deck of 52. Find the probability of a face card.

Face cards: J, Q, K in 4 suits → 12
P = 12/52

Answer: 3/13 ≈ 0.2308

Example 3: P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. Find P(A or B).

P(A or B) = 0.5 + 0.4 − 0.2

Answer: 0.7

Outcomes, events and the sample space

An experiment (tossing a coin, throwing a die) has outcomes; the list of all outcomes is the sample space. An event is a set of outcomes you are interested in — “an even number” on a die is {2, 4, 6}. When all outcomes are equally likely, the probability of the event is the number of its outcomes divided by the total: 3/6 = 1/2.

With two dice there are 36 equally likely outcomes, not 11 sums — a sum of 7 can happen 6 ways but a sum of 2 only one way, which is why 7 is the most likely sum.

Independent vs mutually exclusive

Mutually exclusive events cannot happen together (getting a 2 and a 5 on one throw), so you add their probabilities. Independent events do not affect each other (a head on the coin and a 6 on the die), so you multiply. Two events with non-zero probabilities cannot be both at once — if one happening rules out the other, they clearly affect each other.

🎯 Asked in competitive exams

This topic regularly appears in the quantitative aptitude section of:

IBPS PO & Clerk
SBI PO
SSC CGL
Campus placement aptitude tests

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Frequently asked questions

What is the formula of probability?

P(E) = number of outcomes favourable to E ÷ total number of equally likely outcomes. It is always between 0 and 1 (0% to 100%).

What is the probability of getting a head when a coin is tossed?

1/2 — there are 2 equally likely outcomes (H, T) and 1 is a head. For two coins, P(at least one head) = 3/4.

How many face cards are there in a deck of 52?

12 — the jack, queen and king of each of the 4 suits. So P(face card) = 12/52 = 3/13. There are 26 red cards, 4 aces and 13 cards of each suit.

What is the difference between P(A and B) and P(A or B)?

P(A and B) is the chance that both happen; P(A or B) is the chance that at least one happens. P(A or B) = P(A) + P(B) − P(A and B).

Can a probability be more than 1 or negative?

No. 0 means impossible and 1 means certain. If you get a value outside 0 to 1, recheck the counting. The calculator rejects such inputs and impossible combinations.