Probability

Probability quantifies the chance or likelihood of an event occurring. It is a number between 0 and 1 (or 0% to 100%), where:

• 0 means the event never occurs
• 1 means the event always occurs
• Values between 0 and 1 show the degree of likelihood

Example: Tossing a fair coin has two equally likely outcomes: Heads or Tails. Probability of getting Heads = $\frac{1}{2} = 0.5$.

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2. Key Formulas & Shortcuts

Let:

• $S$ = Sample space (total possible outcomes)
• $E$ = Event (favorable outcomes)
• $n(S)$ = Number of elements in sample space
• $n(E)$ = Number of elements favorable to event

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A. Probability of an event $E$:

$$P(E) = \frac{n(E)}{n(S)}$$

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B. Probability of complement of $E$ (event not happening):

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

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C. Addition Rule (for two events $A$ and $B$):

• If mutually exclusive (cannot occur together):

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

• If not mutually exclusive:

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

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D. Multiplication Rule:

• For independent events $A$ and $B$:

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

• For dependent events, use conditional probability (covered below).

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E. Conditional Probability:

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

Probability of $A$ given $B$ has occurred.

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F. Probability of "At least one":

$$P(\text{at least one}) = 1 - P(\text{none})$$

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G. Total Probability (Law of Total Probability):

If $B_1, B_2, ..., B_n$ form a partition of the sample space,

$$P(A) = \sum_{i=1}^n P(A|B_i) P(B_i)$$

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3. Conceptual Tips & Common Mistakes

• Always check if outcomes are equally likely before using simple ratio formula.

• Don't confuse union and intersection:

  • Union $A \cup B$ = event $A$ or $B$ or both.
  • Intersection $A \cap B$ = event $A$ and $B$ both.

• Conditional probability requires dividing by the probability of the given condition/event.

• For "at least one" problems, using complement saves time and effort.

• Make sure events are independent before applying multiplication rule directly.

• Draw sample spaces or tree diagrams for complex problems.

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5. Solved Examples

Example 1: Basic Probability

Q: A die is rolled once. What is the probability of getting an even number?

A:

• Sample space, $n(S) = 6$ (1, 2, 3, 4, 5, 6)
• Favorable outcomes, $n(E) = 3$ (2, 4, 6)

$$P(E) = \frac{3}{6} = \frac{1}{2}$$

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Example 2: Complement Rule

Q: Probability of not getting a 6 in one roll of a die?

A:

$$P(\text{not }6) = 1 - P(6) = 1 - \frac{1}{6} = \frac{5}{6}$$

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Example 3: Union of Two Events

Q: From a deck of cards, probability of drawing a card that is a King or a Heart?

A:

• $P(King) = \frac{4}{52}$
• $P(Heart) = \frac{13}{52}$
• $P(King \cap Heart) = \frac{1}{52}$ (King of Hearts)

$$P(K \cup H) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$$

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Example 4: Independent Events

Q: Two coins tossed. Probability both show heads?

A:

$$P(H) = \frac{1}{2} \quad\Rightarrow\quad P(HH) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$$

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Example 5: Conditional Probability

Q: A bag has 3 red and 2 blue balls. One ball is drawn. Given that it is red, what is the probability that the next ball drawn is also red (without replacement)?

A:

After 1 red ball is taken, remaining balls = 4, red balls = 2

$$P(\text{2nd red} | \text{1st red}) = \frac{2}{4} = \frac{1}{2}$$

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Example 6: At Least One

Q: Probability of getting at least one head in two tosses of a coin?

A:

$$P(\text{no heads}) = P(TT) = \frac{1}{4}$$ $$P(\text{at least one head}) = 1 - \frac{1}{4} = \frac{3}{4}$$