Probability
Probability at this level is counting favourable outcomes over total outcomes. The whole subject rests on being able to count both correctly, which is a listing skill more than a formula skill.
Mental model
Probability is bookkeeping: count the favourable outcomes, count the total outcomes, divide. So the real skill is COUNTING correctly, and the first question to ask is always 'does order matter?' — if yes it is a permutation, if no it is a combination. Second question: are the events independent (multiply) or mutually exclusive (add)? Third: is 'at least one' involved? Then compute 1 minus the probability of none, which is almost always the shorter path.
Associations
A short trigger on the left, everything it pulls with it on the right.
Probability = favourable / total, and total must count every equally likely outcome.
For two coins, the total outcomes are HH, HT, TH, TT — four, not three — because HT and TH are distinct even though both 'look like' one head and one tail. Listing outcomes explicitly, rather than reasoning about them abstractly, avoids this exact undercounting error.
'Without replacement' shrinks the total for the second draw.
Drawing a second card from a reduced deck (without replacement) changes the total outcomes for that draw — a common exam trick is stating 'without replacement' in the problem and expecting you to still use the original total for both draws.
Things to remember
The night-before list. Short enough to recall cold.
- 1Probability always lies between 0 (impossible) and 1 (certain), inclusive.
- 2For a single fair die, each of the six faces has probability 1/6; the probability of an even number is 3/6 = 1/2.
- 3For two independent events, the probability of both happening is the product of their individual probabilities.
- 4The probability of an event NOT happening is 1 minus the probability of it happening.
In detail
- Probability of an event = favourable outcomes / total outcomes, and always lies between 0 and 1 inclusive.
- P(not A) = 1 - P(A); for 'at least one', compute 1 minus the probability that none occurs.
- A standard deck has 52 cards: 4 suits of 13, 26 red and 26 black, 12 face cards, and 4 aces.
- Two dice give 36 equally likely outcomes; a sum of 7 is the most likely, occurring in 6 of them.
- For independent events P(A and B) = P(A) x P(B); for mutually exclusive events P(A or B) = P(A) + P(B). Treating overlapping events as mutually exclusive is the standard error.
Where people slip
- For two coin tosses, do not count only three outcomes (two heads, two tails, one of each) — HT and TH are distinct, giving four equally likely outcomes in total.
Check yourself
Nothing here is scored — it will not change your mastery figures.
Two coins are tossed together. What is the probability of getting exactly one head?
From the question bank
A die is thrown once. Find the probability of getting a 4.
From the question bank
A coin is tossed once. Find the probability of getting a head.
Books to read
- NCERT Class X — Mathematics
- NCERT Class XI — Mathematics
- Quantitative Aptitude for Competitive Examinations — R S Aggarwal
Official sources
Hand-checked official domains only — the notes above carry no links of their own.