Probability — Grade X

Grade X

PROBABILITY || सम्भाव्यता

Probability सम्भाव्यता

Math — Grade X · Chapter 8

📚 3 Topics

🎲 Additive · Multiplicative · Tree Diagram

🔢 Mixed Exercise included

📚

Probability

Click any topic to expand subtopics

Exercise — Model 1

Show

📐

Key Formulas

Probability — essential laws & formulas

🌳 Tree Diagram — Coin tossed twice

┌─ H ½

├─ HH ¼

└─ HT ¼

└─ T ½

├─ TH ¼

└─ TT ¼

Basic Probability

$$P(A) = \frac{\text{Favourable outcomes}}{\text{Total outcomes}}$$

Always $0 \leq P(A) \leq 1$. Certain event: $P=1$. Impossible event: $P=0$.

Complementary Event

$$P(A') = 1 - P(A)$$

$A'$ (or $\bar{A}$) is the event that $A$ does NOT occur. $P(A) + P(A') = 1$.

Additive Law — General

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

For any two events $A$ and $B$. Subtract the intersection to avoid double-counting.

Additive Law — Mutually Exclusive

$$P(A \cup B) = P(A) + P(B)$$

When $A$ and $B$ cannot both occur: $P(A \cap B) = 0$. E.g. rolling a 3 or a 5.

Multiplicative Law — Independent

$$P(A \cap B) = P(A) \times P(B)$$

When $A$ and $B$ are independent — the outcome of one does not affect the other.

Multiplicative Law — Dependent

$$P(A \cap B) = P(A) \times P(B|A)$$

$P(B|A)$ = probability of $B$ given $A$ has already occurred (conditional probability).

Conditional Probability

$$P(B|A) = \frac{P(A \cap B)}{P(A)}$$

The probability of $B$ occurring, knowing that $A$ has already occurred.

Tree Diagram — Branch Product

$$P(\text{outcome}) = \prod \text{branch probabilities}$$

Multiply along branches to get the probability of a combined outcome.

Tree Diagram — Sum of All Ends

$$\sum_{\text{all paths}} P(\text{path}) = 1$$

All end-branch probabilities must sum to 1 — use this to check your tree.

Exhaustive & Equally Likely

$$P(S) = 1, \quad P(\emptyset) = 0$$

$S$ = sample space (all outcomes). $\emptyset$ = empty set (impossible event).

Odds in Favour

$$\text{Odds} = \frac{P(A)}{P(A')} = \frac{P(A)}{1-P(A)}$$

Expresses probability as a ratio of favourable to unfavourable outcomes.

💡

Key Concepts

Understand beyond memorising formulas

🎲

Basic Probability

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). Always list the sample space first. Count favourable outcomes carefully — drawing a Venn diagram or table helps avoid errors.

🔄

Complementary Events

The complement $A'$ means "not A". Since either $A$ or $A'$ must happen, $P(A)+P(A')=1$. This is one of the most useful shortcuts: it is often easier to find $P(A')$ and subtract from 1 than to count favourable outcomes directly.

➕

Additive Law

For "A or B", use $P(A \cup B) = P(A)+P(B)-P(A \cap B)$. If events are mutually exclusive (cannot happen together), the intersection is zero so you just add. Always check first: can both happen at the same time?

✖️

Multiplicative Law

For "A and B", multiply probabilities. If independent: $P(A \cap B)=P(A)\times P(B)$. If dependent (e.g. drawing without replacement): use $P(A)\times P(B|A)$ where the second probability changes after the first event.

🔀

Independent vs Dependent

Events are independent if one does not affect the other (e.g. tossing two coins, rolling two dice). Events are dependent if the first changes the conditions for the second (e.g. drawing cards without replacement).

🌳

Tree Diagram

Draw branches for each stage. Write probabilities on branches — each set of branches from one point must sum to 1. To find the probability of a path, multiply along branches. To find "at least one" outcomes, sum the relevant end-paths.

⭕

Mutually Exclusive Events

Two events are mutually exclusive if they cannot both occur in the same trial. E.g. getting a Head and a Tail on a single coin toss. Their intersection is empty: $P(A \cap B)=0$, so $P(A \cup B) = P(A)+P(B)$.

📋

Problem-Solving Strategy

Step 1: Identify the sample space. Step 2: Determine if events are independent/dependent and mutually exclusive or not. Step 3: Choose the correct law (additive or multiplicative). Step 4: For multi-stage problems, always draw a tree diagram.

📋