Probability

Probability

Concept Explanation

Probability is the branch of mathematics that quantifies how likely an event is to occur. Think of it as a way of assigning a number between 0 and 1 to any outcome — 0 means it will never happen, 1 means it’s guaranteed. When you flip a fair coin, the probability of heads is exactly 0.5, meaning it’s equally likely to happen as not happen. This framework shows up everywhere: weather forecasts, insurance premiums, medical test results, and of course, exam questions.

When we’re interested in more than one event happening, things get more interesting. If you want to know the probability that it rains today AND you remember your umbrella, you can’t just multiply individual probabilities unless the events are independent — meaning one doesn’t affect the other. If those events aren’t independent, you need to account for how one changes the likelihood of the other using conditional probability.

The most common mistake students make is double-counting. If event A and event B can both happen, simply adding P(A) + P(B) will count the situation where both occur twice. The inclusion-exclusion principle fixes this by subtracting the overlap. Always ask yourself: can these events happen at the same time?

Key Formulas

Symbol	Meaning
P(A)	Probability that event A occurs
P(A∪B)	Probability that A or B or both occur
P(A∩B)	Probability that both A and B occur
P(B|A)	Probability of B given that A has already occurred
P(not A)	Probability that A does not occur

Step-by-Step Example

Q: A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. What is the probability that both are red?

Step 1: Find P(first ball is red) = 5/8

Step 2: After removing one red ball, 4 red and 3 blue remain. P(second ball is red | first was red) = 4/7

Step 3: Multiply: P(both red) = (5/8) × (4/7) = 20/56 = 5/14

Answer: 5/14

Common Mistakes

• Assuming independence when there’s no replacement → Drawing without replacement means events are dependent; multiply sequentially instead
• Forgetting to subtract the intersection in P(A∪B) → Always apply the inclusion-exclusion formula: P(A∪B) = P(A) + P(B) − P(A∩B)

Quick Test (2 Qs)

1. Q: Two dice are thrown. What is P(sum equals 8)? Options: A) 1/6 B) 5/36 C) 7/36 D) 1/9. Ans: B) 5/36 (Reason: Favorable pairs are (2,6),(3,5),(4,4),(5,3),(6,2) = 5 out of 36 total outcomes)
2. Q: A card is drawn from a standard deck. P(face card or heart)? Options: A) 4/13 B) 11/26 C) 3/13 D) 1/2. Ans: A) 4/13 (Reason: P(face)=12/52, P(heart)=13/52, P(both)=3/52. So 12/52+13/52−3/52=22/52=11/26… Wait — actually 12+13−3=22, 22/52=11/26. Answer is B. But let me recalculate: 12 face cards + 13 hearts = 25, but 3 face cards are hearts (J,Q,K of hearts), so 25−3=22. 22/52=11/26. Hmm the options don’t quite match — let me fix: Answer should be 11/26 ≈ B.)