The base formula
P = favorable outcomes / total outcomes, with every outcome equally likely. List or count both sides carefully.
One die, P(even) = 3/6 = 1/2.
Rapid Revision · Quantitative Aptitude
Probability
Probability is counting twice: favorable outcomes over total outcomes. Most placement questions use coins, dice, cards and balls, so learn those sample spaces cold.
The 3-minute recap
If you read nothing else tonight, read these 6 lines.
- P(event) = favorable / total; always between 0 and 1.
- P(not A) = 1 - P(A); 'at least one' = 1 - P(none).
- Independent events multiply; mutually exclusive events add.
- Deck: 52 cards, 4 suits x 13, 12 face cards, 26 red and 26 black.
- Two dice: 36 outcomes; sum 7 is the most likely (6 ways).
- Without replacement: shrink the denominator each draw.
Formula sheet
Every formula for probability in one place, each labelled so you know exactly when to reach for it. Screenshot it the night before.
Basic probability
P(E) = favourable outcomes / total outcomes
Always 0 <= P(E) <= 1.
Complement
P(not E) = 1 - P(E)
For 'at least one', 1 - P(none) is almost always faster.
Addition (OR)
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 (AND)
P(A and B) = P(A) x P(B)
Conditional
P(A | B) = P(A and B) / P(B)
Work through the cards
7 cards, each one idea: what it is, a worked example, and the trap to dodge.
At least one
Compute the complement: P(at least one) = 1 - P(none). Almost always faster than adding cases.
3 coin tosses, at least one head: 1 - (1/2)^3 = 7/8.
Trap: Summing P(exactly 1) + P(exactly 2) + ... invites slips; use 1 - P(none).
Add or multiply?
A AND B for independent events: multiply. A OR B for mutually exclusive events: add. General OR: P(A) + P(B) - P(A and B).
Die then coin, P(6 and heads) = 1/6 x 1/2 = 1/12.
Trap: Independent and mutually exclusive are different things; exclusive events are never independent.
Cards you must know
52 cards: 4 suits of 13; 26 red (hearts, diamonds), 26 black; face cards J, Q, K in each suit = 12 total; 4 aces.
P(face card) = 12/52 = 3/13. P(red king) = 2/52 = 1/26.
Trap: An ace is usually NOT counted as a face card.
Two dice
36 equally likely pairs. Count ways for each sum: sum 2 has 1 way, sum 7 has 6, sum 12 has 1.
P(sum 9) = 4/36 = 1/9 (3+6, 4+5, 5+4, 6+3).
Without replacement
Each draw changes the pool: multiply stage probabilities with updated numerator and denominator.
Bag 4 red + 3 blue, two draws, both red: 4/7 x 3/6 = 2/7.
Trap: Keeping the denominator at 7 for the second draw.
Odds vs probability
Odds in favor a:b convert to probability a/(a+b). Odds against a:b means P = b/(a+b).
Odds in favor 3:2 means P = 3/5.
Go deeper
A recap is not practice. These are the creators we rate for real depth on probability; full credit to each.
One topic down. Keep the streak going.
Each recap takes 3 minutes; the full set covers everything the first round tests. And when the test is cleared, your resume takes the next screen.
Original content by OptiResume; facts and formulas are common knowledge, the wording is ours. Go-deeper links go to creators we rate; we are not affiliated with them.