Probability
PROBABILITY
Introduction to Probability
| Concept | Meaning | Example |
|---|---|---|
| Chance | Possibility of an event happening | Chance of rain tomorrow |
| Probability | Numerical measure of chance | Probability of rain = 0.6 |
Probability always lies between 0 and 1.
Probability does not predict the future, it only measures chances based on known outcomes.
Chance and Probability in Real Life
| Situation | Event | Possible Outcome |
|---|---|---|
| Tossing a coin | Head or Tail | 50% chance each |
| Rain forecast | Rain / No Rain | Given as % probability |
Probability helps in decision-making under uncertainty.
In sports, weather prediction, insurance – probability helps estimate risks.
Impossible and Certain Events
| Type | Probability Value | Example |
|---|---|---|
| Impossible Event | 0 | Getting number 7 on a die |
| Certain Event | 1 | Sun rises in the east |
Impossible Event → Probability = 0
Certain Event → Probability = 1
Certain Event → Probability = 1
Events with probability between 0 and 1 are called uncertain events.
Basic Concepts of Probability
| Term | Meaning |
|---|---|
| Experiment | An action with uncertain result |
| Outcome | Result of an experiment |
| Sample Space | Set of all outcomes |
| Event | Subset of sample space |
Probability = Favorable outcomes ÷ Total outcomes
Always write the sample space before calculating probability.
Finding Probability
| Experiment | Sample Space | Probability |
|---|---|---|
| Throwing a die | {1,2,3,4,5,6} | P(number < 3) = 2/6 = 1/3 |
| Tossing a coin | {H, T} | P(Head) = 1/2 |
Total probability of all possible outcomes = 1
Always reduce probability to simplest fraction form.
Complementary Events
| Event | Complement | Relation |
|---|---|---|
| A | Not A | P(A) + P(Not A) = 1 |
Complementary events sum always equals 1.
If P(A) is known, then P(Not A) = 1 − P(A).
Mutually Exclusive Events
| Definition | Condition | Example |
|---|---|---|
| Events that cannot occur together | No common outcome | Getting Head or Tail in one toss |
For mutually exclusive events A and B:
P(A or B) = P(A) + P(B)
P(A or B) = P(A) + P(B)
If two events overlap, they are NOT mutually exclusive.
Probability with Playing Cards
| Fact | Value |
|---|---|
| Total cards in a deck | 52 |
| Hearts / Diamonds / Clubs / Spades | 13 each |
| Red cards | 26 |
| Black cards | 26 |
P(Heart) = 13/52 = 1/4
P(Red card) = 26/52 = 1/2
P(Red card) = 26/52 = 1/2
Face cards = Jack, Queen, King (Total 12).
Applications of Probability
| Field | Application |
|---|---|
| Weather | Rain forecast |
| Sports | Winning chances |
| Insurance | Risk calculation |
| Exams | Guessing-based probability |
Probability helps in risk analysis and decision-making.
In competitive exams, questions often link probability with real life scenarios like dice, cards, or balls from bags.