Probabilistic Propositions

A field of mathematics dealing with the examination of random processes is known as probability theory. A random event’s outcome cannot be predicted before it occurs, although it could be any of multiple possibilities. The final result is said to be dictated by chance.

In everyday speech, the word probability has numerous meanings. Two of them are particularly crucial for the advancement and application of probability theory in mathematics. One is the representation of probability as relative frequencies, which may be demonstrated using basic games such as coins, cards, dice, & roulette wheels. The unique aspect of games of chance would be that the outcome of a single trial cannot be anticipated with certainty, even if the aggregate outcomes of several trials show some pattern.

As per the relative frequency interpretation, the statement that the probability of “heads” in tossing a coin equals one-half implies that the relative frequency with which “heads” actually occurs will be approximately one-half in a large number of tosses, even though it contains no implication regarding the outcome of any given toss. 

Many other instances exist, involving groupings of individuals, gas molecules, genes, and so forth. Actuarial statements on life expectancy for people over a certain age represent the collective experience of a huge number of people, but they do not claim to predict what will happen to any one individual.

Predictions concerning the likelihood of a genetic condition arising in a child of parents with established genetic makeup are similar in that they are assertions about relative probabilities of occurrence in a vast number of cases, not predictions about a specific individual.

1. P(E) = [Number of favourable outcomes of E]/ [total number of possible outcomes of E] is the probability of an occurrence E.
2. A sure event or certain event has a probability of 1.
3. The chance of an impossible occurrence occurring is nil.
4. The probability of an occurrence E is a number P(E) with a value of 0 ≤ P ≤ (E) and a value of 1 P (E). It’s always a positive number when it comes to probability.
5. Let A & B are two events which are mutually exclusive, then P(A⋃B) = P(A) + P(B).
6. A single-outcome event is referred to as an elementary event. P(A) + P(A’) = 1. The sum of the probability of such events in an experiment is 1.
7. The sum of the probabilities of an event and its complementary event is 1.
8. P(A⋃B) = P(A) + P(B) – P(A⋂B).
9. P(A⋂B) = P(A) + P(B) – P(A⋃B).
10. If A1, A2, A3 ………, An are mutually exclusive events, then P (A1 ⋃ A2 ⋃ A3… ⋃ An) = P(A1) + P (A2) + ………. + P(An)

Examples:

Example 1: When tossing a die, calculate the chances of getting an even number.

S = {1, 2, 3, 4, 5, 6} is the solution.

Favourable circumstances = {2,4,6}

The number of favourable events equals three.

There are a total of 6 results.

As a result, the probability is P = 3/6 = 1/2.

Example 2: In a box of 200 watches, 190 are in good condition, seven have minor flaws, and three have serious flaws. Reena, a merchant, will only accept watches that are in good condition, but Seema, another trader, would only reject watches that have significant flaws. From the carton, one watch is selected at random. How likely is it that Reena will accept it? Determine the likelihood that Seema will accept it.

Solution:

From a pool of 200 timepieces, one is selected at random. As a result, there are 200 possible outcomes, each of which is equally plausible.

Reena’s number of acceptable outcomes = 190/200 = 0.95

No. of Seema-acceptable outcomes = (190+7)/200 = 197/200 = 0.9

Gambling has always been associated with probability theory, and many of the most accessible examples still derive from that practice. The basic instruments of the gambling trade should be familiar to you: a coin, a (six-sided) die, as well as a full deck of 52 cards. Heads are the result of a fair coin toss.

A fair die would give you 1, 2, 3, 4, 5, or 6 with equal probability if you roll heads (H) or tails (T).

With a shuffled deck of cards, any arrangement of cards is equally likely.

Equally likely outcomes:

Assume you do an experiment with n alternative outcomes, totalling a set S. Assume that all possibilities are equally likely. (Whether or not this assumption is reasonable depends on the circumstances.) Example 1.2 shows an example where this is not a realistic assumption.)

E S is a set of outcomes for an event E. If an event E has m alternative outcomes (commonly referred to as “excellent” outcomes for E), then P(E) = m/n is the probability of E.

Conclusion

The key point would be that the premises of a valid argument can be uncertain, in which case the (un)certainty of the conclusion is not constrained by (deductive) validity. The reasoning with premises ‘if it rains tomorrow, I will get wet’ and ‘it will rain tomorrow’ and conclusion ‘I will get wet’ is valid, but if the second claim is uncertain, the conclusion will typically be doubtful as well. 

Propositional probability logics explore how uncertainties, such as probabilities, ‘flow’ from the premises to the conclusion; in other words, they study probability preservation rather than truth preservation.