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Probability

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1. Basic Definition

Probability is the measure of the likelihood of an event occurring.

$P(E) = \frac{\text{Number of Favourable Outcomes}}{\text{Total Number of Possible Outcomes}}$ or $P(E) = \frac{n(E)}{n(S)}$

Where $n(S)$ is the Sample Space.

Example:

Q: Find the probability of getting a Head when tossing a coin.

Solution: Total Outcomes (

S

) = {H, T}

\to n(S)=2

.
Favourable (

E

) = {H}

\to n(E)=1

P(H) = 1/2

2. Limits of Probability

The probability of an event lies between 0 and 1 (inclusive).

$0 \le P(E) \le 1$

$P(E) = 0$ : Impossible Event (e.g., Rolling a 7 on a standard die).
$P(E) = 1$ : Certain Event (e.g., Sun rising in the East).
$P(E) + P(\text{not } E) = 1$ (Complementary Events).

3. Coins Concepts

1 Coin: {H, T} $\to 2^1 = 2$ .
2 Coins: {HH, HT, TH, TT} $\to 2^2 = 4$ .
3 Coins: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} $\to 2^3 = 8$ .

Trick: For 'At least 1 Head', use

1 - P(\text{No Head})

Example:

Q: 3 coins are tossed. Prob of getting at least 1 Head?

Solution: Total = 8. No Head (TTT) = 1 case.

P(\text{At least 1 H}) = 1 - 1/8 = 7/8

4. Dice Concepts

1 Die: {1, 2, 3, 4, 5, 6} $\to 6^1 = 6$ .
2 Dice: (1,1) to (6,6) $\to 6^2 = 36$ .

Sum on 2 Dice (Shortcut):
Min Sum 2 (1 way), Max Sum 12 (1 way).
Sum 7 has max ways (6 ways).
Pattern: 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 (for sums 2 to 12).

5. Playing Cards

Total Cards = 52.

Suits (13 each): Spades (♠), Clubs (♣) [Black]; Hearts (♥), Diamonds (♦) [Red].
Face Cards: King, Queen, Jack (3 per suit $\times$ 4 = 12 total).
Honours: A, K, Q, J.

6. Balls & Marbles (AND/OR Rules)

AND Rule (Multiplication): Event A AND Event B occur $\to P(A) \times P(B)$ . (Independent events).
OR Rule (Addition): Event A OR Event B occur $\to P(A) + P(B)$ (Mutually Exclusive).

Selection: Use Combinations (

{}^{n}C_{r}

P(\text{Selection}) = \frac{\text{Favourable Combinations}}{\text{Total Combinations}}

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Probability

1. Basic Definition

2. Limits of Probability

3. Coins Concepts

4. Dice Concepts

5. Playing Cards

6. Balls & Marbles (AND/OR Rules)

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