1.2 Notation and Terminology
- 1.
- A random experiment is a process that ends in exactly one of several possible outcomes, which one not being known in advance. Tossing a coin and rolling a die are random experiments.
- 2.
- An outcome is a possible result of the experiment, for example getting a \(4\) when a die is rolled. Exactly one outcome occurs on each trial.
- 3.
- The sample space \(S\) is the set of all possible outcomes. For one roll of a die, \(S = \{1, 2, 3, 4, 5, 6\}\).
- 4.
- An event is a subset of the sample space. For example \(E = \{2, 4, 6\}\) is the event that the score is even. The event \(E\) occurs if the outcome lies in \(E\).
- 5.
- Events \(A_1, A_2, \dots , A_n\) are mutually exclusive if \(A_i\cap A_j = \emptyset \) whenever \(i\ne j\), that is, no two can occur together.
- 6.
- Outcomes are equally likely if each has the same chance of occurring.
Independence of events, which is a different notion from mutual exclusiveness, is defined in section 1.4.
Example 1.2.2. Write down the sample space for each experiment.
- 1.
- Two coins are tossed and the face showing on each is recorded.
- 2.
- Two coins are tossed and the number of heads is recorded.
- 3.
- A lot of \(N\) items contains \(D\le N\) defectives. Items are drawn one at a time, without replacement, until the last defective is found, and the number of items drawn is recorded.
- 4.
- A light bulb is run until it fails, and its lifetime is recorded.
Solution.
- 1.
- \(S = \{HH, HT, TH, TT\}\), where \(H\) means head up and \(T\) tail up.
- 2.
- \(S = \{0, 1, 2\}\).
- 3.
- At least \(D\) items must be drawn and at most \(N\), so \(S = \{D, D + 1, \dots , N\}\).
- 4.
- \(S = \{t : t\ge 0\}\), where \(t\) is the lifetime.
The same physical experiment can have different sample spaces, depending on what is recorded: parts 1 and 2 describe the same tosses.
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