Basics of Probability
Basics of Probability
Probability closes the mathematics half of Part-III and contributes 1 to 2 questions. At class 8 level it stays with coins, dice and cards, so the whole topic reduces to counting outcomes correctly and remembering that probability can never exceed one.
The Definition
P(E) = Number of favourable outcomes ÷ Total number of outcomes
This applies only when all outcomes are equally likely, which is why questions specify a fair coin or an unbiased die.
- Probability always lies between 0 and 1 inclusive.
- An impossible event has probability 0; a certain event has probability 1.
- The sum of the probabilities of all possible outcomes is exactly 1.
- P(not E) = 1 − P(E), which is the fastest route to many answers.
✗ The probability of an event can be 1.5 or −0.2 | ✓ Probability is never below 0 or above 1
Sample Spaces to Know
| Experiment | Total outcomes |
|---|---|
| One coin | 2 |
| Two coins | 4 |
| Three coins | 8 |
| One die | 6 |
| Two dice | 36 |
| Pack of cards | 52 |
For n coins the count is 2ⁿ. For two dice it is 36, and the number of ways of getting a given sum is what most dice questions ask. A sum of 7 has six ways, the most of any sum, while 2 and 12 have one each.
The Pack of Cards
- 52 cards in four suits of 13.
- Red suits: hearts and diamonds, 26 cards. Black suits: spades and clubs, 26 cards.
- Face cards: jack, queen and king — 12 in all, 3 per suit.
- Aces are not face cards, though many candidates count them as such.
- There are 4 of each rank, so 4 aces, 4 kings and so on.
The ace question is the standard trap: asked for the probability of drawing a face card, a candidate who counts 16 instead of 12 gets a ready-made wrong option.
Worked Examples
- Probability of an even number on a die: 3 favourable out of 6, so 1/2.
- Probability of at least one head on two coins: the outcomes are HH, HT, TH, TT, so 3/4. Alternatively 1 − P(no head) = 1 − 1/4.
- Probability of a red king: there are 2 red kings out of 52, so 1/26.
- Probability of a sum of 7 with two dice: 6 favourable out of 36, so 1/6.
The at least one phrasing is worth attention: it is almost always faster to compute the complement and subtract from one, and the paper sets those questions expecting exactly that move.
कम से कम एक वाले प्रश्न में उलटा निकालकर 1 में से घटाईए — यही तेज रास्ता है।
TRE pointer: Two items decide this topic: the ace is not a face card rule, and the complement method for at-least-one questions. Both have a wrong option built from the obvious mistake. Because probability sits at the end of the class 8 syllabus, papers keep it simple, which makes these one or two marks among the most reliable in the whole Part-III paper. With −1/3 negative marking, count the sample space on paper rather than from memory.
60-Second Recap
- P(E) = favourable ÷ total, for equally likely outcomes only.
- Probability lies between 0 and 1; impossible is 0, certain is 1.
- P(not E) = 1 − P(E) — use it for at-least-one questions.
- n coins give 2ⁿ outcomes; two dice give 36; a pack has 52 cards.
- 12 face cards, not 16 — aces are not face cards.
- A sum of 7 is the most likely with two dice, with 6 ways out of 36.
Frequently Asked Questions
Is an ace a face card?
No. Face cards are the jack, queen and king only, giving twelve in a pack of fifty-two. Counting the four aces as well gives sixteen, and that wrong figure is always offered among the options in questions asking for the probability of drawing a face card.
How do I solve at-least-one probability questions?
Find the probability that the event does not happen at all and subtract it from one. For at least one head in two tosses, the probability of no head is one quarter, so the answer is three quarters. This is faster and safer than listing favourable outcomes.
Can probability be greater than one?
No. Probability always lies between zero and one inclusive, because the favourable outcomes can never exceed the total outcomes. A value of zero means the event is impossible and a value of one means it is certain. Values outside this range are placed as distractors.
How many ways can a sum of seven appear with two dice?
Six ways, out of thirty-six total outcomes, giving a probability of one sixth. Seven has more ways than any other sum, while two and twelve have just one each. Listing the pairs on paper takes ten seconds and avoids a counting error.
Next step: Geometry and mensuration decide the largest block of the Part-III maths paper. Test this chapter in the BPSC TRE 4.0 mock test on ExamAtlas — five options, −1/3 negative marking, and questions set at the depth this chapter prepares you for.