We have learnt in previous chapter about Experiments, Sample Space and Events. One important information about the events is the chance/probability of it's occurance.

Probability Function¶

Let's start our discussion by looking at the conditions that such a probability function must satisfy?

Axioms of Probability¶

✅ The first Axiom defines the probability to be non negative i.e it should be greater than or equal to zero.

✅ The second Axiom says that the probability of sample space is 1.

✅ Third Axiom defines the probability of an event can be computed as the sum of the probability of the disjoint outcomes contained in the event.

All probability functions must follow the Axioms of probability and once they follow the Axioms, certain property gets associated with it. Let's look at some of the Properties of Probability

Properties of Probability Function¶

Property 1 : $$ P(A) = 1 - P(A^c) $$

Proof : $$ A ∪ A^c = Ω $$

$$ P(A ∪ A^c ) = P(A) +P(A^c) $$ $$ 1 = P(A) +P(A^c) $$

$$\therefore P(A) = 1 - P(A^c) $$

Property 2 : $$P(A) ≤ 1$$

Proof : $$ \because P(A) = 1 - P(A^c) $$ $$As, \: P(A^c) ≥ 0$$

$$so, \: P(A) ≤ 1$$

Property 3 : $$P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$$

Proof : $$ P(A ∪ B) = P(A ∪ (B ∩ A^c))$$ $$ = P(A) + P(B ∩ A^c)$$ $$ = P(A) + P(B) - P(B ∩ A)$$

$$ \because P(B ∩ A^c) = P(B) - P(B ∩ A)$$

As, $$ P(B) = P((B ∩ A^c) ∪ (B ∩ A))$$

$$ P(B) = P(B ∩ A^c) + P(B ∩ A)$$ $$ P(B ∩ A^c) = P(B) - P(B ∩ A)$$

Property 4 : The sum of the probabilities of all outcomes is equal to 1 $$Ω = A_1 ∪ A_2 ∪ .... ∪ A_n$$

Property 5 : $$P(Φ) = 0$$ $$P(Ω) = P(Ω ∪ Φ) = P(Ω) + P(Φ) $$ $$⇒ 1 = 1 + P(\phi)$$ $$⇒ P(\phi) = 0$$

Defining Probability Functions¶

As Relative Frequency¶

Goal : Assign a number to each event such that this number reflects the chance of the experiment resulting in that event

Thus we have to come up with a function to which when we pass an event, it will assign a number that reflects the chance of experiment resulting in that event.

Required : The probability function must satisfy the axioms of probability

Thus Relative frequency could be defined as :

$P(A_i) = \frac{no.\: of \:times \:the \:outcome\: is\: in \:A_i}{total\: no.\: of\: times\: the\: experiment\: was\: repeated}$

Suppose $A_1, A_2, ... A_n $ are the outcomes, then every event is a union of these outcomes

If the frequencies of $A_1, A_2.... A_n $ are known then the probability of any event can be computed

Although there are $A_1, A_2.... A_n $ outcomes. The number of events is 2^n and every event is a union of these outcomes. Thus the Axioms are more about events then outcomes.

As equally likely outcomes¶

For an equally likely outcome

$$P(H) = P(T)=k$$ $$Ω = H ∪ T$$ $$P(Ω) = P(H ∪ T)$$ $$= P(H) + P(T)$$ $$= 2k$$

$$Thus, \: \: P(H) = P(T) = k = \frac{1}{2}$$

Thus for equally likely outcomes, If $A_i$ : event that the outcome is i

then, $$A_1, A_2, A_3, A_4, A_5, A_6 \: partitions\: Ω$$

$$P(A_1) = P(A_2) = P(A_3) = P(A_4) = P(A_5) = P(A_6) = k$$

$$P(Ω) = ∑_{i=1}^6 P(A_i) = 6k =1$$

❓Can we derive a formula for computing the probability of events of an experiment with n equally likely outcomes?

Let E : any event with k outcomes (the outcomes are of course disjoint)

then, $$P(E) = ∑_{i=1}^k \frac{1}{n} = \frac{k}{n}$$

finally, $$ P(X) = \frac{number\: of \: outcomes \: in \: X}{number \: of \: outcomes \: in \: Ω} $$

❓ Are the Axioms of Probability Satisfied?

$1. \: P(A_i) ≥ 0 \: $: Since it is a ratio of two positive number

$2. P(Ω) = 1? $ : $Ω$ contains all outcomes

$3. P(A_1 ∪ A_2) = P(A_1)+P(A_2)?:$ $ P(A_1 ∪ A_2) = \frac{k_1 + k_2}{n} = \frac{k1}{n} + \frac{k2}{n} = P(A_1) + P(A_2)$

Examples :