Basic Probability Formula
Probability is the ratio of favourable outcomes to the total number of equally likely possible outcomes in a random experiment:
P(Event) = Favourable Outcomes ÷ Total Possible Outcomes
Combined Event Probability
When two or more events occur together, the calculation depends on whether they are independent or mutually exclusive:
Independent Events (AND probability): Both events must occur. Multiply the individual probabilities.
P(A and B) = P(A) × P(B)
Example: Drawing two cards (with replacement) and getting hearts both times:
P = 13/52 × 13/52 = 0.25 × 0.25 = 0.0625 (6.25%)
Mutually Exclusive Events (OR probability): At least one of the events occurs. The events cannot happen simultaneously.
P(A or B) = P(A) + P(B)
Example: Rolling a 2 or a 5 on a die:
P = 1/6 + 1/6 = 2/6 = 1/3 (33.3%)
Non-Mutually Exclusive Events (OR with overlap): Events can both occur simultaneously. Subtract the overlap to avoid double-counting.
P(A or B) = P(A) + P(B) − P(A and B)
Example: Drawing a red card OR a king from a deck:
P(red) = 26/52, P(king) = 4/52, P(red king) = 2/52
P = 26/52 + 4/52 − 2/52 = 28/52 ≈ 53.8%
P(A and B) = P(A) × P(B) [independent]
P(A or B) = P(A) + P(B) [mutually exclusive]
P(A or B) = P(A) + P(B) − P(A∩B) [non-exclusive]
Theoretical vs Experimental Probability
There are two approaches to determining probability:
Theoretical Probability: Based on mathematical reasoning about equally likely outcomes. A fair coin has a theoretical probability of 0.5 for heads — determined by logic, not experiment. Requires that all outcomes are equally likely.
Experimental Probability: Based on actual observed outcomes from an experiment or data. Flip a coin 100 times and get 47 heads — experimental probability = 47/100 = 0.47.
As the number of trials increases, experimental probability tends toward theoretical probability. This is called the Law of Large Numbers. In 10 coin flips, getting 7 heads is not unusual. In 10,000 flips, getting 7,000 heads would be extraordinary.
Probability in Real Life
Understanding probability improves decision-making in many practical contexts:
Insurance: Premiums are priced based on the probability of a claim. A 1-in-1,000 chance of a ₹10 lakh claim justifies a premium of around ₹1,000 (plus insurer's costs and margin).
Medical Testing: A test with 95% sensitivity (true positive rate) and 99% specificity (true negative rate) still produces many false positives when testing a rare disease (1-in-1,000 prevalence). Understanding conditional probability helps interpret medical results.
Financial Markets: Options pricing (Black-Scholes model) is rooted in probability. Expected value calculations guide portfolio diversification.
Games and Gambling: The "house edge" in casino games is a probability calculation showing the long-term mathematical advantage the house holds. In European roulette, the house edge is 2.7% — meaning for every ₹100 wagered, the player expects to lose ₹2.70 on average over many spins.