Definitions [7]
Random Experiment: An experiment in which all possible results are known in advance, but the result of any particular performance cannot be predicted with certainty.
Key Features
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All possible outcomes are known before the experiment.
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The exact outcome of one trial cannot be predicted.
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The experiment can usually be repeated under similar conditions.
Outcome: The result of a random experiment
Important: An outcome is a single possible result, not the complete list of results.
Equally Likely Outcomes: Outcomes are equally likely when there is no reason to expect one outcome more often than another.
Conditions
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The coin, die, cards, or selection method should be fair or unbiased.
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Each outcome should have the same chance of occurring.
Sample Space: The set of all possible outcomes of a random experiment. It is usually denoted by S.
Sample Point: Each element of the sample space is called a sample point.
Number of Sample Points: If S is a sample space, then n(S) denotes the number of sample points in S.
The outcomes satisfying a particular condition are called favourable outcomes.
A set of favourable outcomes of a given sample space is an ‘event’. Event is a subset of the sample space.
- Events are denoted by capital letters A, B, C, D etc.
- The number of elements in event A is written n(A).
- Since A ⊆ S, always 0 ≤ n(A) ≤ n(S).
- Example: two coins tossed, A = getting at least one tail → A = {TT, TH, HT}, n(A) = 3.
In Mathematical language, when possibility of an expected event is expressed in number, it is called ‘Probability’. It is expressed as a fraction or percentage using the following formula.
\[ P(A) = \frac{\text{Number of sample points in event } A}{\text{Number of sample points in sample spaces}} = \frac{n(A)}{n(S)} \]
- The Probability is expressed as a fraction or a percentage.
- The probability of any event is from \[ 0 \text{ to } 1 \text{ or } 0\% \text{ to } 100\%. \]
- If \[ E \] is any event, \[ 0 \leq P(E) \leq 1 \text{ or } 0\% \leq P(E) \leq 100\%. \]
Key Points
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A random experiment has known possible outcomes but an unpredictable result.
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An outcome is one possible result of an experiment.
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Outcomes are equally likely when each has the same chance.
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The sample space S is the set of all possible outcomes.
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A sample point is one element of the sample space.
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n(S) is the number of outcomes in the sample space.
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One coin: n(S) = 2-; two coins: n(S) = 4; three coins: n(S) = 8.
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One die: n(S) = 6; two dice: n(S) = 36.
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A standard card pack has 52 cards, 4 suits, 12 face cards, and 4 aces.
- The sample space \[ S \] contains all possible outcomes.
- An event is a subset of the sample space.
- A simple event has one outcome; a compound event has more than one outcome.
- A certain event has probability \[ 1; \] an impossible event has probability \[ 0. \]
- The complement of \[ A \] is \[ A', \] and \[ P(A') = 1 - P(A). \]
- Mutually exclusive events cannot occur together: \[ A \cap B = \varnothing. \]
- Exhaustive events together cover the entire sample space: \[ A \cup B = S. \]
- For equally likely outcomes, \[ P(A) = \frac{n(A)}{n(S)}. \]
- For any two events, \[ P(A \cup B) = P(A) + P(B) - P(A \cap B). \]
- Probability = possibility of an expected event expressed in number.
- \[ P(A) = n(A) / n(S). \]
- Expressed as a fraction or a percentage.
- \[ 0 \leq P(E) \leq 1, \] i.e. \[ 0\% \text{ to } 100\%. \]
