PROBABILITY
CONTENT
(a) Definitions and examples of: (i) Experimental outcomes (ii) Random experiment (iii) Sample space (iv) Sample points (v) Event space (vi) Probability.
(b) Practical example of each term.
(c) Theoretical Probability.
(d) Equiprobable sample space; Definition, Unbiasedness.
(e) Simple probable on equiprobable sample space.
SAMPLE SPACE: Any result of an experiment in probability is usually called an outcome. If we cannot predict before hand, the outcome of an experiment, the experiment is called a random experiment.
The set of all possible outcomes of any random experiment will be called a sample space and it will denoted by \(S\). The number of outcomes in \(S\) or the number of elements in the sample space will be denoted \(n(S)\).
EVENT SPACE: A subset of the sample space which may be a collection of outcomes of a random experiment is called an event space. We shall denote an event space by \(E\), and the number of outcomes or elements in \(E\) by \(N(E)\).
The probability of an event \(E\) denoted \(Pr(E)\) is defined as \(Pr(E)= \frac{n(E)}{n(S)}\)
Since the empty set \(θ\) is a subset of the sample space, \(n(θ) = 0\\\)
\(Pr(θ)= \frac{n(θ)}{n(S)} = 0\) or \(Prob(S)= \frac{n(S)}{n(S)} = 1\)
Example 1: In a single throw of a fair coin, find the probability that:
(i) a head appears
(ii) a tail appears
Solution
Let \(S\) be the sample space, then
\(S = \{H, T\} \\ n(S) = 2\)
Let \(E_1\) be the event that a head appears, then
\(E_1 = \{H\} \\ n(E_1) = 1 \\ Prob.(E_1) = \frac{n(E_1)}{n(S)} = \frac{1}{2}\)
Let \(E_2\) be the event that a tail appears, then
\(E_2 = \{T\} \\ n(E_2) = 1 \\ Prob(E_2) = \frac{n(E_2)}{n(S)} = \frac{1}{2}\)
Example 2: In a single throw of two fair coins, find the probability that:
(a) two heads appears
(b) two tails appears
(c) one head and one tail appears
Solution:
Let S be the sample space then,
\(S = [HH, TT, HT, T] \\ n(S) = 4 \)
(a) Let \(E_1\) be the event that two heads appears, then
\(E_1 = \{ HH\}, n(E_1) = 1 \\ Prob(E_1) = \frac{n(E_1)}{n(S)} = \frac{1}{4} \)
(b) Let \(E_2\) be the event that two tails appears, then
\(E_2 = \{ TT\}, n(E_2) = 1 \\ Prob(E_2) = \frac{n(E_2)}{n(S)} = \frac{1}{4} \)
(c) Let \(E_3\) be the event that one head and one tail appear, then
\(E_3 = \{ TH, HT\}, n(E_3) = 2 \\ Prob(E_3) = \frac{n(E_3)}{n(S)} = \frac{2}{4} = \frac{1}{2} \)
PROPERTIES OF PROBABILITY
The following are some fundamental properties of probability for finite sample space.
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