Probability is the measure of uncertainty of various phenomenons, numerically. It can have positive value from 0 to 1.
The words ‘probably’, ‘doubt’, ‘most probably’, ‘chances’, etc., used in the statements above involve an element of uncertainty.
Concept of Probability is used in insurance industry to calculate the premium; it is also used in stock markets, gambling industry, weather forecasting, manufacturing industry, Science Experiments etc.
Probability started with Gambling industry. In 1654, a gambler Chevalier de Mere, approached the 17th century French philosopher and mathematician Blaise Pascal regarding certain dice problems.
Pascal became interested in these problems, studied them and discussed them with another French mathematician, Pierre de Fermat. Both Pascal and Fermat solved the problems independently. This work was the beginning of Probability Theory.
In case of Empirical or Statistical or Experimental approach to Probability, we perform experiment & find the probability of occurrence of an event based on experimental data.
P(E) = Number of trials in which the event happened / Total number of trials
The requirement of repeating an experiment has some limitations, as it may be very expensive or unfeasible in many situations.
E.g. Repeating the experiment of launching a satellite in order to compute the empirical probability of its failure during launching, or the repetition of the phenomenon of an earthquake to compute the empirical probability of a multistoried building getting destroyed in an earthquake.
Also the results of the experimental approach are not consistent. Thus we use Classical Approach to Probability.
In classical approach we assume Probability of certain events. We know, in advance, that the coin can only land in one of two possible ways — either head up or tail up. We can reasonably assume that each outcome, head or tail, is as likely to occur as the other. We refer to this by saying that the outcomes head and tail, are equally likely.
Thus p(Head) = p(tail) = ½
For another example of equally likely outcomes, suppose we throw a die once. For us, a die will always mean a fair die. What are the possible outcomes? They are 1, 2, 3, 4, 5, 6. Each number has the same possibility of showing up. So the equally likely outcomes of throwing a die are 1, 2, 3, 4, 5 and 6.
Therefore p(1) = p(2) = p(3) = p(4) = p(5) = p(6) = 1/6
In this chapter we will assume that all the experiments have equally likely outcomes.
The theoretical probability (also called classical probability) of an event E, written as P(E), is defined as
P(E) = Number of outcomes favourable to E /Number of all possible outcomes of the experiment
where we assume that the outcomes of the experiment are equally likely.
Sum of the probabilities of all the elementary events of an experiment is 1
0 ≤ P(E) ≤ 1
Numerical: A die is thrown once. Find the probability of getting (i) a prime number; (ii) a number lying between 2 and 6; (iii) an odd number
Solution: In a throw of a die, possible outcomes are 1, 2, 3,4, 5,6
Primes numbers in this set are : 2, 3,5
P(prime) = Number of outcomes for Prime/ Total number of outcomes = 3/6 = ½
Number lying between 2 & 6 are : 3,4, 5
P(Number b/w 2 & 6) = Number of outcomes for number between 2 & 6/ Total number of outcomes = 3/6 = ½
Odd numbers in the set are: 1, 3, 5
P(Odd Number) = Number of outcomes for odd number / Total number of outcomes = 3/6 = ½
Numerical: One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) a king of red colour (ii) a face card (iii) a red face card (iv) the jack of hearts (v) a spade (vi) the queen of diamonds
Solution: Total number of cards is 52
Number of King of red color card = 2
P(King of red colour) = Number of outcomes for king of red Colour / Total number of outcomes
Or P(King of red colour) = 2/52 = 1/26
Number of face card is 12
P(face card) = Number of outcomes for face card / Total number of outcomes
Or P(face card) = 12/52 = 3/13
Number of Red face card is 12
P(Red face card) = Number of outcomes for Red face card / Total number of outcomes
Or P(Red face card) = 6/52 = 3/26
Number of Jack of Heart is 1
P(Jack of Hearts) = Number of outcomes for jack of Hearts / Total number of outcomes
Or P(Red face card) = 1/52
Number of Spade Card is 13
P(Spade Card) = Number of outcomes Spade Card / Total number of outcomes
Or P(Red face card) = 13/52 = ¼
Number of Queen of Diamonds Card is 1
P(Queen of Diamonds) = Number of outcomes for Queen of Diamonds / Total number of outcomes
Or P(Red face card) = 1/52
Impossible & Sure Events
Probability of an event which is impossible to occur is 0. Such an event is called an impossible event.
Probability of an event which is sure (or certain) to occur is 1. Such an event is called a sure event or a certain event
Numerical: A jar contains lemon only. Malini takes out one fruit without looking into the bag. What is the probability that she takes out an apple? Also find probability of taking out a lemon?
Solution: Since jar has only lemons, Probability of talking out Lemon is 1, since it is sure that that item she will take out will be lemon.