Definitions [3]
Probability measures the degree of certainty of the occurrence of an event.
The conditional probability of both events A and B over the sample space S is
Two events are said to be independent if the occurrence of one does not depend on the other.
For two events E and F:
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E and F are independent if P(F | E) = P(F), when \[P(E) \neq 0\].
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Similarly, E and F are independent if P(E | F) = P(E), when \[P(F) \neq 0\].
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An equivalent and most commonly used test is:
Formulae [1]
\[P(A | B) = \frac{P(A \cap B)}{P(B)}\], where \[P(B) \neq 0\].
\[P(B | A) = \frac{P(A \cap B)}{P(A)}\], where \[P(A) \neq 0\].
Multiplication rule: \[P(A \cap B) = P(B) \cdot P(A | B) = P(A) \cdot P(B | A)\].
Complement form: \[P(A' | B) = 1 - P(A | B)\].
Theorems and Laws [1]
For two events:
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\[P(E \cap F) = P(F) \cdot P(E | F)\]
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\[P(E \cap F) = P(E) \cdot P(F | E)\]
For three events:
- \[P(E \cap F \cap G) = P(E) \cdot P(F | E) \cdot P(G | E \cap F)\]
Key Points
| No. | Term | Definition |
|---|---|---|
| 1 | Probability | A measure of the chance of occurrence of an event. |
| 2 | Random Experiment | An experiment in which all possible outcomes are known, but the exact outcome cannot be predicted with certainty. |
| 3 | Outcome | The result of a random experiment. |
| 4 | Sample Space (S) | The set of all possible outcomes of a random experiment. |
| 5 | Sample Point | Each element of the sample space. |
| 6 | Number of Sample Points | The number of elements in the sample space is denoted by n(S). |
| 7 | Equally Likely Outcomes | Outcomes which have the same chance of occurring. |
| No. | Term | Definition |
|---|---|---|
| 1 | Probability | A measure of the chance of occurrence of an event. |
| 2 | Random Experiment | An experiment in which all possible outcomes are known, but the exact outcome cannot be predicted with certainty. |
| 3 | Outcome | The result of a random experiment. |
| 4 | Sample Space (S) | The set of all possible outcomes of a random experiment. |
| 5 | Sample Point | Each element of the sample space. |
| 6 | Number of Sample Points | The number of elements in the sample space is denoted by n(S). |
| 7 | Equally Likely Outcomes | Outcomes which have the same chance of occurring. |
Playing Cards – Key Facts
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Total cards = 52
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Red cards = 26 (Hearts, Diamonds)
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Black cards = 26 (Clubs, Spades)
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Each suit has 13 cards
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Face cards = King, Queen, Jack (Total = 12)
| Type of Event | Meaning | Probability |
|---|---|---|
| Sure (Certain) Event | An event that is certain to occur | P(E) = 1 |
| Impossible Event | An event that cannot occur | P(E) = 0 |
| Simple (Elementary) Event | An event having only one outcome | P(E) = 1 / n(S) |
| Complementary Event (E̅) | An event that occurs when E does not occur | P(not E) = 1 − P(E) |
| Mutually Exclusive Events | Two events that cannot occur together | P(A ∩ B) = 0 |
| Exhaustive Events | Events which together cover all outcomes of S | P(A₁) + P(A₂) + … = 1 |
| Equally Likely Events | All outcomes have the same chance of occurring | P(E) = n(E) / n(S) |
| General Rule | Probability of any event | 0 ≤ P(E) ≤ 1 |
Properties:
- Complement Rule
P(A′) = 1 − P(A)
⇒ P(A) + P(A′) = 1 - Range of Probability
0 ≤ P(A) ≤ 1 - Impossible Event
P(ϕ) = 0 - Certain Event
P(S) = 1 - Subset Rule
If A ⊆ B, then P(A) ≤ P(B) - Difference of Events
P(A ∩ B′) = P(A) − P(A ∩ B)
P(A′ ∩ B) = P(B) − P(A ∩ B) - Union of Two Events
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) - Union of Three Events
P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
− P(A ∩ B) − P(B ∩ C) − P(C ∩ A) + P(A ∩ B ∩ C) - Mutually Exclusive Events (2 events)
If A ∩ B = 0, then
P(A ∪ B) = P(A) + P(B) - Mutually Exclusive Events (multiple)
P(A₁ ∪ A₂ ∪ ... ∪ Aₙ) = P(A₁) + P(A₂) + ... + P(Aₙ) - Upper Bound of Union
P(A ∪ B) ≤ P(A) + P(B)
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Conditional probability means probability under a given condition.
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The formula is \[P(A | B) = \frac{P(A \cap B)}{P(B)}\], where \[P(B) \neq 0\].
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Always reduce the sample space according to the condition first.
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The numerator represents outcomes common to both events.
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Do not confuse P(A | B) with P(B | A).
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For independent events, P(A | B) = P(A).
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Multiplication theorem is used to find the probability of simultaneous occurrence of events.
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For two events: \[P(E \cap F) = P(E) \cdot P(F | E)\]
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Another equivalent form is \[P(E \cap F) = P(F) \cdot P(E | F)\].
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For three events: \[P(E \cap F \cap G) = P(E) \cdot P(F | E) \cdot P(G | E \cap F)\].
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Most “without replacement” questions are solved using this theorem.
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Always define events before solving a probability problem.
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Independent events do not influence each other.
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The standard test is \[P(E \cap F) = P(E)P(F)\].
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Conditional form: \[P(F | E) = P(F)\] and \[P(E | F) = P(E)\], when defined.
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If two events are independent, related complement pairs are also independent.
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Mutually exclusive events and independent events are different.
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For independent events A and B, \[P(A \cup B) = 1 - P(A')P(B')\].
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For three events, mutual independence requires pairwise conditions and the condition involving all three together.
