What is Probability?
Probability measures how likely an event is to happen. It is expressed as a number between 0 (impossible) and 1 (certain).
Colored Balls Bag Model
Imagine a bag containing 3 Red balls and 2 Blue balls (Total 5 balls).
Favorable Outcomes (Red Balls) = 3
Total Possible Outcomes = 5
$P(\text{Red}) = \frac{3}{5} = 0.6 = 60\%$
Probability is a high-value topic in CAT, NMAT, SNAP, and XAT. It combines core ideas from Permutations, Combinations, Counting, and Logical Reasoning. Once you count favorable and total outcomes correctly, probability questions become straightforward.
Connected Modern Math Modules:
Probability of an event occurring equals 1 minus probability of the event NOT occurring.
If event A occurring has no impact on event B, multiply their probabilities.
Probability of event A OR event B occurring (subtracting overlap).
Rolling a Fair Die
Q: A fair 6-sided die is rolled once. Find the probability of getting a number greater than 4.
Step 1: Total sample space $S = \{1, 2, 3, 4, 5, 6\} \implies n(S) = 6$.
Step 2: Favorable outcomes for $> 4$ are $\{5, 6\} \implies n(E) = 2$.
Step 3: $P(E) = \frac{n(E)}{n(S)} = \frac{2}{6} = \mathbf{\frac{1}{3}}$.
Drawing Cards from a Standard Deck
Q: A card is drawn at random from a standard 52-card deck. Find the probability of drawing a King.
Step 1: Total cards in a deck $n(S) = 52$.
Step 2: Total Kings in a deck (Spade, Heart, Diamond, Club) $n(E) = 4$.
Step 3: $P(\text{King}) = \frac{4}{52} = \mathbf{\frac{1}{13}}$.
"At Least One" Coin Toss with Complement Rule
Q: Two fair coins are tossed simultaneously. Find the probability of getting at least one Head.
Step 1 (Sample Space): $S = \{HH, HT, TH, TT\} \implies n(S) = 4$.
Step 2 (Complement Trick): The opposite of "at least one Head" is "NO Heads" (which is TT).
Step 3: $P(\text{No Heads}) = P(TT) = \frac{1}{4}$.
Step 4 (Apply Complement Rule): $P(\text{At least one Head}) = 1 - P(\text{No Heads}) = 1 - \frac{1}{4} = \mathbf{\frac{3}{4} = 0.75}$.
❌ Trap 1: Wrong Denominator
Using Favorable in Denominator
The denominator MUST always be the total number of possible outcomes $n(S)$, not the remaining unfavorable outcomes.
❌ Trap 2: Unequal Likelihood
Assuming Outcomes are Equally Likely
Before using $P = \frac{n(E)}{n(S)}$, ensure all sample outcomes are equally likely (e.g. fair coin, unbiased die).
❌ Trap 3: Ignoring Complement
Counting "At Least One" Directly
Counting multiple favorable cases directly takes time and causes missing case errors. Always use $1 - P(\text{None})$.
⚡ The "At Least One" Complement Shortcut
Instead of adding $P(1) + P(2) + P(3) + \dots$, calculate $1 - P(0)$. Saves up to 2 minutes on CAT questions!
⚡ Combinations for Ball/Selection Drawings
When selecting $r$ items simultaneously from a pool of $n$, use combinations for numerator & denominator.
Question 1 (Basic):
A bag contains 6 red, 4 blue, and 5 green balls. If one ball is drawn at random, find the probability of drawing a green ball.
Solution: Total balls $n(S) = 6 + 4 + 5 = 15$.
Favorable green balls $n(E) = 5$.
$\text{Probability } P = \frac{5}{15} = \mathbf{\frac{1}{3}}$.
Question 2 (Moderate):
Two fair dice are rolled simultaneously. Find the probability that the sum of the numbers shown is 8.
Solution: Total sample space when rolling 2 dice $n(S) = 6 \times 6 = 36$.
Favorable pairs for sum = 8 are: $(2,6), (3,5), (4,4), (5,3), (6,2) \implies n(E) = 5$.
$\text{Probability } P = \mathbf{\frac{5}{36}}$.
Q: Can probability ever be greater than 1?
No. Probability is always a real number between 0 and 1 inclusive ($0 \le P(E) \le 1$). 0 indicates an impossible event and 1 indicates a certain event.
Q: Why do we use combinations in probability?
Because most selection-based probability questions (drawing balls from a bag, choosing committee members) involve selecting items where the order of selection does not matter.
Q: When should I use the Complement Rule in probability?
Use the Complement Rule $P(E) = 1 - P(E')$ whenever a question asks for "at least one", "not", or "none". Calculating the opposite event is almost always much faster.