Definitions [2]
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
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]
If B1, B2,..., Bn are mutually exclusive and exhaustive events and if A is an event consequent to these Bi's, then for each i = 1, 2, 3, ..., n,
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)
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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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Bayes' Theorem works from effect to cause.
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Prior probability means “before observation”.
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Posterior probability means “after observation”.
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The observed event is usually given in the question statement.
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The required answer is often a probability of the form P(cause ∣ observed event).
