Axiomatic Approach to probability

Function: If  A and B are two non-empty sets, then a function from A to B, denoted as f(.): A→ B is a rule, which associates to each element x∊A to a unique element y∊B. The element y is called the image of x and written as y=f(x) is called a pre-image of y. The set is called the domain of f(.) and  B its codomain or counter domain.

𝝈 Algebra: Let X be a non-empty set and àbe a collection of a subset of X. Then Ã is called a sigma-algebra or 𝝈-algebra if the following properties are satisfied

1.  X ∈ Ã 

2. If A ∈ Ã  then Ā ∈ Ã

3. if <Aₙ> is a sequence of a subset in Ã, then ∪ₙ₌₁ Aₙ = à

Sample Space: It is the collection of all possible outcomes of a random experiment. It is denoted by Ω. For example:

1. When we toss a coin, then Ω.={H, T}
2. When we toss two coins at a time, then Ω={HH, HT, TH, TT}
3. When we throw a die, then Ω={1,2,3,4, 5, 6}

Event: An event is a subset of the sample space Ω of a random experiment. For example, When we toss two coins at a time, then Ω={HH, HT, TH, TT}, and A={HH, TT}   is an event.

Event space: The collection of all events of a random experiment is called the event space. We denote an event space by the letters Ã,   etc. If the sample space Ω is finite consisting of  n points then the event space consists of  2ⁿ events. For example

1. If Ω={H, T} then event space Ã={φ, {H},{T}, {H, T}}

Probability function: Let Ω denote the sample space and Ã a 𝝈-algebra of events (each a subset of Ω). A probability space denoted as P[.], is defined as a function  which satisfies the following axioms:

        P1: P[A] ≥ 0 for each A ∈ Ã 

        P2: P[Ω] = 1.

        P3:  P[  Aⱼ] = ∑ Aⱼ Where <Aⱼ> is a sequence of mutually exclusive events in Ã.

Probability Space: If Ω is a sample space, Ã is a 𝝈-algebra of events (each a subset of Ω) and P[.] is a probability function with domain Ã, then the triplet (Ω, Ã, P[.]) is called a probability space.

Theorem: Prove the following:

1.  P[φ]=0.
2. P[  Aⱼ] = ∑ Aⱼ Where <Aⱼ> is a sequence of mutually exclusive events in Ã.
3.  For any event A ∈ Ã, P[Ā ]=1-P[A].
4. For any two events A & B ∈ Ã,   P[A ⋃ B]= P[A]+ P[B] - P[A በ B].
5. For any two events A & B ∈ Ã,  and A ⊂ B, then P[A] ≤ P[B]

Example 1. If P[A] = $\frac{1}{3}$ and $P[\bar{B}]=\frac{1}{4}$, can $A$ and $B$ be disjoint?

Solution: Suppose A and B are disjoint events. Then

$$ A\cap B = \phi \Rightarrow A \subset \bar{B} \Rightarrow P[A] \leq P[\bar{B}]\Rightarrow \frac{1}{3} \leq \frac{1}{4}$$

Which is impossible. Hence $A$ and $B$ cannot be disjoint.

Example 2. Three newspapers are published in a city and a survey of readers indicates the following:

20% read A, 16% read B, 14% read C.

8% read both A and B, 5% read both A and C.

4% read both B and C, 2% read all the three.

For a person chosen at random, find the probability that he reads none of the paper. $c^2$

Solution: We have $P[A]=20\% =\frac{2}{100}, P[B]= 16\%=\frac{16}{100}, P[C]=14\% =\frac{14}{100}$

$P[A\cap B]=\frac{8}{100}, P[A\cap C]=\frac{5}{100}, P[B\cap C]=\frac{4}{100}, P[A \cap B \cap C]=\frac{2}{100} $ 

The probability that a person reads A or B or C is given by

$P[A\cup B \cup C]= P[A] + P[B] + P[C] - P[A \cap B]- P[A \cap C] - P[B \cap C]+P[A\cap B \cap C]$

$\Rightarrow P[A\cup B \cup C] = \frac{20}{100}+\frac{16}{100}+\frac{14}{100}-\frac{8}{100}-\frac{5}{100}-\frac{4}{100}+\frac{2}{100}=\frac{7}{100}$

Hence the probability that he reads none of the papers

$$ P[\overline{A\cup B\cup C}]=1-P[A\cup B\cup C]=1-\frac{7}{20}=\frac{13}{20}$$.

Example 3. One urn contains three red balls, two white balls and one blue ball. A second urn contains one red ball, two white balls and three blue balls.

1. Describe the sample space for this experiment if one ball is selected at random from each urn.
2. If the balls in the two urns are mixed in a single urn and then a sample of three is drawn. Find the probability that all the three colors are represented when sampling is done (a) with replacement (b) without replacement.

Solution: Urn $I = \{r_1, r_2, r_3, w_1, w_2, b\},$ Urn $II=\{R, W_1, W_2, B_1, B_2, B_3\}$

Here $r_i, R$ denote red color; $w_i, W_1$ white color and $b, B_i$ blue color.

1. Each urn contains $6$ balls. The sample space of the experiments consists of $6 \times 6 = 36$ points. 

$$ \Omega=\{\lbrace r_1, r_2, r_3,w_1, w_2, b\rbrace \times \lbrace R, W_1, W_2, B_1, B_2, B_3\rbrace\}$$

        2.       If the balls in the two urns are mixed in a single urn, then the latter urn contains 12 balls out of which 4 balls are red, 4 white and 4 blue. Draw a sample of 3 balls, where the sampling is done with replacement.

The probability of the first ball (of any color) $=\frac{12}{12}$

Now we have 8 balls out of the remaining two color and probability  of the second ball (of the second color) = $\frac{8}{12}$

Lastly the probability of the third ball (out of the third color) $=\frac{4}{12}$.

Hence the probability that a sample of 3 balls contains all the three colors is $=\frac{12}{12} \times \frac{8}{12} \times \frac{4}{12}=\frac{2}{9}$

When the sampling is done without replacement, the probability that a sample of three balls contains all the three color is  

$$ = \frac{4C_1 \times 4C_1 \times 4C_1}{12C_3}=\frac{4 \times 4 \times 4 \times 3 \times 2}{12 \times 11 \times 10}=\frac{16}{55}$$

Exercises:

1. If $P[\bar{A}]=\alpha $ and $P[\bar{B}]=\beta,$ then prove that $P[AB]=1-\alpha-\beta$

2. If $B \subset A ,$ then $P[A\cap \bar{B}]=P[A]-P[B]$

3. Find the probability that a certain 13-cards hand contains exactly 8 spades.

4.  One urn contains three red balls, two white balls and one blue ball. A second urn contains one red ball, two white balls and three blue balls. One ball is selected at random from each urn

                a) Find the probability that both balls will be of the same color.

                b) Is the probability that both balls will be red greater than the probability that both will be white?

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