Probability | SuccessSetu

1️⃣ Random Experiment

An experiment which gives different outcomes every time.

Experiment → Outcome → Sample Space → Probability

Example: Tossing a coin → H, T

2️⃣ Sample Space

All possible outcomes of an experiment.

S = {H, T}

Two coins → {HH, HT, TH, TT}

3️⃣ Probability Formula

P(E) = Favorable / Total Outcomes

Die → P(2) = 1/6

4️⃣ Types of Events

➤ Impossible → P = 0
➤ Sure Event → P = 1
➤ Complement → 1 – P(E)

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1. Impossible & Sure Event

Impossible: P(A) = 0

Sure: P(A) = 1

Example: Die → Getting 7 (Impossible), 1–6 (Sure)

2. Simple Event

Only one outcome

Sample Space = {1,2,3,4,5,6}

Example: Getting 2

3. Compound Event

More than one outcome

{2,4,6}

Even number on die

4. Mutually Exclusive Events

Cannot occur together

P(A∩B) = 0

P(A∪B) = P(A)+P(B)

2 and 5 on die

5. Non-Mutually Exclusive

Can occur together

P(A∪B)=P(A)+P(B)-P(A∩B)

Even & Prime

6. Independent Events

One does not affect other

P(A∩B)=P(A)P(B)

Coin + Die

7. Dependent Events

One affects other

P(A∩B)=P(A)P(B|A)

Picking balls

8. Complementary Event

Opposite event

P(A') = 1 − P(A)

Head / Tail

If P(Rain)=0.3 → No Rain=0.7

9. Equally Likely Events

Same probability

Each = 1/6

Die outcomes

10. Exhaustive Events

Cover all outcomes

{1,2,3,4,5,6}

Odd + Even

📝 PYQ – Question 1

Two coins are tossed. Find probability of one head.

a) 1/4
b) 1/2
c) 3/4
d) 1

Show Answer

✔ Answer: 1/2

📝 PYQ – Question 2

A die is thrown. Find probability of prime number.

a) 1/6
b) 1/2
c) 1/3
d) 2/3

Show Answer

✔ Answer: 1/2

📝 PYQ – Question 3

Probability value lies between?

a) -1 to 1
b) 0 to 1
c) 1 to ∞
d) None

Show Answer

✔ Answer: 0 to 1

📌 Important Note

✔ Always write Sample Space
✔ Count favorable cases
✔ Apply formula

🎯 Concept 1 : Coin Toss

When a coin is tossed, possible outcomes are:

Sample Space = {H, T}

Toss Coin → H / T → Probability

Example: Probability of Head = 1/2

🎯 Concept 2 : Dice Throw

When a dice is thrown, outcomes are:

{1, 2, 3, 4, 5, 6}

Throw Dice → 1–6 → Result

Example: P(6) = 1/6

🎯 PYQ Practice

Q1. A coin is tossed twice. Find P(2 Heads)?

A) 1/4
B) 1/2
C) 3/4
D) 1

Show Answer

✅ Answer: A (1/4)

🎯 Formula Section

P(A) = Favourable / Total Outcomes

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

🎯 Flow Chart

Question → Sample Space → Favourable → Probability → Answer

📌 Coin Toss

One Coin → H, T

Two Coins → HH, HT, TH, TT

Outcomes = 2ⁿ

Example: 3 coins → 2³ = 8

S = {HH, HT, TH, TT} Total Outcomes = 4
P(HH) = 1/4

📌 Two Dice

Total Outcomes:

6 × 6 = 36 Total = 6ⁿ

Example: 2 Dice → 36

📌 Addition Formula

For two events A and B:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Used in overlapping events

📌 Cards Probability

Total Cards = 52

P(Heart) = 13/52 = 1/4

Same for Spade, Diamond, Club

📌 Important Points

0 ≤ P(E) ≤ 1
Total Probability = 1
Sample space is necessary

🔹 Negation of an Event

Not A = P(A̅)

A or B (Union) = A ∪ B

A and B (Intersection) = A ∩ B

A but not B = A ∩ B̅

Exactly one of A or B = (A ∩ B̅) ∪ (A̅ ∩ B)

🔹 Addition Theorems

Theorem 1: Mutually Exclusive
(Nothing common: A ∩ B = Ø)
P(A ∪ B) = P(A) + P(B)

Theorem 2: Non-Mutually Exclusive
(Something is common: A ∩ B ≠ 0)
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

🔹 3-Set Venn Diagram

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(C ∩ A) + P(A ∩ B ∩ C)

All three = A ∩ B ∩ C
Exactly two = b + d + e
Exactly one = a + f + g

🔹 Key Results

Sum of all probabilities = 1

At least two = b + d + e + c

At most two = 1 - P(A ∩ B ∩ C)

1️⃣ Random Experiment

2️⃣ Sample Space

3️⃣ Probability Formula

4️⃣ Types of Events

1. Impossible & Sure Event

2. Simple Event

3. Compound Event

4. Mutually Exclusive Events

5. Non-Mutually Exclusive

6. Independent Events

7. Dependent Events

8. Complementary Event

9. Equally Likely Events

10. Exhaustive Events

📝 PYQ – Question 1

📝 PYQ – Question 2

📝 PYQ – Question 3

📌 Important Note

🎯 Concept 1 : Coin Toss

🎯 Concept 2 : Dice Throw

🎯 PYQ Practice

🎯 Formula Section

🎯 Flow Chart

📌 Coin Toss

📌 Two Dice

📌 Addition Formula

📌 Cards Probability

📌 Important Points

📝 Practice Questions (PYQ)

🔹 Negation of an Event

🔹 Addition Theorems

🔹 3-Set Venn Diagram

🔹 Key Results