Probability Basics

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For SBI Clerk Prelims the whole topic rests on one formula — favourable outcomes divided by total outcomes — applied to coins, dice and cards. There is no calculus and no deep theory; the skill is counting the two numbers correctly. Once you can list the sample space for the standard experiments, every basic probability question becomes a short fraction you can simplify in seconds.

The Core Formula and Sample Space

Probability of an event equals the number of favourable outcomes over the total number of possible outcomes, provided all outcomes are equally likely. The total set of outcomes is the sample space, and getting its size right is where most marks are won or lost. A single coin has 2 outcomes, two coins have 4, a die has 6, and a standard deck has 52 cards. Fix these base counts firmly, because nearly every Prelims probability question is built on one of them.

Coins and Dice

For coins, the probability of one head on a single toss is one-half. With two coins the four outcomes are HH, HT, TH and TT, so exactly one head has probability two-fourths, or one-half. For a die, each face has probability one-sixth, and events like an even number or a number greater than four are counted directly from the six faces. With two dice the sample space is thirty-six ordered pairs, and questions about a particular sum are solved by counting the pairs that produce it — a sum of seven, for instance, has six.

Playing Cards

A standard deck has fifty-two cards split into four suits — hearts and diamonds are red, spades and clubs are black — with thirteen cards each. The face cards are the jack, queen and king, giving twelve face cards in all, and there are four aces. So the probability of drawing a king is four over fifty-two, or one-thirteenth, and of drawing a red card is twenty-six over fifty-two, or one-half. Memorising the deck's composition removes all the counting effort from card questions.

P(event) = favourable / total 0 ≤ P ≤ 1 P(not A) = 1 − P(A)

Solved Examples

Example 1: Probability of an even number on a die = 3/6 = 1/2, since 2, 4 and 6 are the three even faces. Example 2: Probability of drawing an ace from a full deck = 4/52 = 1/13, because four of the fifty-two cards are aces — a direct favourable-over-total count.

✗ Counting two coins as having 3 outcomes (0, 1, 2 heads)  |  ✓ Listing all 4 equally likely outcomes HH, HT, TH, TT

The Complement Shortcut

When a question asks for the probability of at least one of something, computing it directly can mean adding several cases, which is slow and error-prone. The shortcut is the complement rule: the probability that an event happens equals one minus the probability that it does not happen. To find the chance of at least one head in three tosses, do not add the one-head, two-head and three-head cases; instead subtract the single no-head case from one. There are eight outcomes for three coins and only one has no head, so the answer is one minus one-eighth, or seven-eighths, in a single step. Any time you see the phrase at least one, reach for the complement first — it collapses a multi-case sum into one quick subtraction and is the most useful probability trick the exam rewards.