Probability

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Probability — Key Facts for NABE (Pakistan)

• P(E) = Favorable Outcomes / Total Outcomes (when outcomes equally likely)
• P(Sure Event) = 1 | P(Impossible Event) = 0
• P(Not A) = 1 - P(A) (Complement Rule)
• P(A or B) = P(A) + P(B) - P(A and B) (Addition Rule)
• ⚡ Exam tip: For independent events, P(A and B) = P(A) × P(B)

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Probability — NABE (Pakistan) Study Guide

Basic Concepts

Experiment: An action with uncertain outcome (e.g., tossing a coin)

Outcome: A single possible result (e.g., “Heads”)

Sample Space (S): Set of all possible outcomes

• Coin toss: S = {H, T}
• Dice roll: S = {1, 2, 3, 4, 5, 6}

Event: A specific outcome or set of outcomes

Probability Scale

0 ≤ P(E) ≤ 1

P(E) = 0: Impossible event
P(E) = 1: Certain event

Basic Probability Formula

For equally likely outcomes:

P(Event) = Number of favorable outcomes / Total number of outcomes

Example: Probability of getting an even number on a die

• Favorable outcomes: {2, 4, 6} = 3
• Total outcomes: {1, 2, 3, 4, 5, 6} = 6
• P(even) = 3/6 = 1/2 = 0.5

Complementary Events

Not A (A’): Event that A does not occur

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

Example: Probability of NOT getting 6 on a die

• P(6) = 1/6
• P(not 6) = 1 - 1/6 = 5/6

NABE Exam Pattern

Common question types:

1. Basic probability of single events
2. Complement rule problems
3. Addition rule (mutually exclusive events)
4. Multiplication rule (independent events)
5. Conditional probability

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Probability — Comprehensive NABE (Pakistan) Notes

Detailed Theory

1. Types of Events

Mutually Exclusive (Disjoint): Two events that cannot occur together

• Example: Rolling a 3 OR a 5 on a die
• If A and B are mutually exclusive: P(A or B) = P(A) + P(B)

Independent Events: The occurrence of one doesn’t affect the other

• Example: Tossing two coins
• If A and B are independent: P(A and B) = P(A) × P(B)

Dependent Events: The occurrence of one affects the other

• Example: Drawing cards without replacement
• P(A and B) = P(A) × P(B|A)

Exhaustive Events: Events that cover all possible outcomes

• Example: Getting odd {1, 3, 5} and even {2, 4, 6} are exhaustive for a die

2. Addition Rule — Venn Diagram Approach

General Formula:

P(A or B) = P(A) + P(B) - P(A and B)

For Mutually Exclusive Events (P(A and B) = 0):

P(A or B) = P(A) + P(B)

Venn Diagram Interpretation:

• Area of A = P(A)
• Area of B = P(B)
• Overlap = P(A and B)
• Union = P(A) + P(B) - P(A and B)

Example: In a class of 40 students, 25 play football, 20 play cricket, and 10 play both.

• P(F) = 25/40 = 5/8
• P(C) = 20/40 = 1/2
• P(F and C) = 10/40 = 1/4
• P(F or C) = 5/8 + 1/2 - 1/4 = (5+4-2)/8 = 7/8

3. Multiplication Rule

For Independent Events:

P(A and B) = P(A) × P(B)

Example: Tossing two coins. Find P(both heads).

• P(H on first) × P(H on second) = 1/2 × 1/2 = 1/4

For Dependent Events:

P(A and B) = P(A) × P(B|A)

Where P(B|A) is probability of B given that A has occurred.

Example: Drawing 2 cards without replacement. Find P(both aces).

• P(first ace) = 4/52
• P(second ace | first ace) = 3/51
• P(both aces) = 4/52 × 3/51 = 12/2652 = 1/221

4. Conditional Probability

Definition: P(B|A) — Probability of B given that A has occurred

P(B|A) = P(A and B) / P(A)

Example: From a deck, P(King | Face Card)?

• P(King and Face) = P(King) = 4/52 = 1/13
• P(Face) = 12/52 = 3/13
• P(King|Face) = (1/13) / (3/13) = 1/3

5. Baye’s Theorem (Advanced)

Statement: P(A|B) = [P(B|A) × P(A)] / P(B)

Example: Medical test problem

• Disease prevalence: 1%
• Test accuracy: 99%
• P(Positive|Disease) = 0.99
• P(Positive|No Disease) = 0.01
• P(Disease|Positive) = (0.99 × 0.01) / [0.99×0.01 + 0.01×0.99] = 0.0099/0.0198 ≈ 50%

6. Permutations and Combinations in Probability

When outcomes are equally likely:

P(E) = n(E) / n(S)

Example: Random 4-digit PIN. Find P(all digits different).

• n(S) = 10^4 = 10,000
• Favorable: 10 × 9 × 8 × 7 = 5040
• P = 5040/10000 = 0.504

7. Odds

Odds in Favor of E = P(E) / P(E’) = n(E) : n(E’)

Odds Against E = P(E’) / P(E) = n(E’) : n(E)

Conversion:

• If odds in favor are a:b, then P(E) = a/(a+b)
• If odds against are c:d, then P(E) = d/(c+d)

8. Expected Value

Definition: Weighted average of all possible values

E(X) = Σ x × P(x)

Example: Game costs Rs. 5 to play. Win Rs. 20 with probability 1/5, otherwise lose. Find expected gain.

• Values: +15 (win - cost) with p=1/5, -5 with p=4/5
• E = 15 × 1/5 + (-5) × 4/5 = 3 - 4 = -1
• On average, lose Rs. 1 per game

9. Probability Distribution

Binomial Distribution (n independent trials):

P(X = r) = nCr × p^r × q^(n-r)

Where p = success probability, q = 1-p

Example: Coin tossed 5 times. Find P(exactly 3 heads).

• n = 5, r = 3, p = 1/2, q = 1/2
• P = 5C3 × (1/2)^3 × (1/2)^2 = 10 × 1/32 = 10/32 = 5/16

10. Common Mistakes to Avoid

1. Non-Equally Likely Outcomes: Don’t assume unless stated
2. Replacement: Check if events are with or without replacement
3. Addition vs. Multiplication: OR usually means addition, AND usually means multiplication
4. Dependent vs. Independent: Check the relationship
5. Complement Use: Sometimes P(A) easier than P(A’), use 1 - P(A’)

Practice Questions for NABE

1. Two dice are rolled. Find probability that sum is 8.
2. A bag contains 5 red and 4 white balls. Two balls are drawn at random. Find P(both red).
3. In a class of 50 students, 30 play cricket and 25 play football. If 10 play both, find P(plays at least one).
4. A card is drawn from a deck. Find P(King or Queen).
5. Three coins are tossed. Find probability of getting at least one head.

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