Mathematics : Probability

Probability theory, a branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.

How likely something is to happen.
Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Probability Formula.

P(A) = n(E)/n(S)

Where,

P(A) is the probability of an event “A”

n(E) is the number of favourable outcomes

n(S) is the total number of events in the sample space

NOTE : Here, the favourable outcome means that the outcome of interest.

Basic  Probability  Formulas.

Probability Range , 0 ≤ P(A) ≤ 1

Rule of Complementary Events ,  P(A’) + P(A) = 1

Examples

Example 1: What is the probability that a card taken from a standard deck, is an Ace?

Solution:  Total number of cards a standard pack contains = 52

A deck of cards contain Ace = 4 cards

So, the number of favourable outcome = 4

Now, by looking at the formula,

Probability of finding an ace from a deck is,

P(Ace) = (Number of favourable outcomes) / (Total number of favourable outcomes)

P(Ace) = 4/52

= 1/13

So we can say that the probability of getting an ace is 1/13.

Example 2: Calculate the probability of getting an odd number if a dice is rolled?

Solution: Sample space (S) = {1, 2, 3, 4, 5, 6}

Let “E” be the event of getting an odd number, E = {1, 3, 5}

So, the Probability of getting an odd number,

P(E) = (Number of outcomes favorable)/(Total number of outcomes)

P(E) = n(E)/n(S) = 3/6 = ½

Question 1 : Complete the following statements

(i) Probability of an event E + Probability of the event ‘not E’ =………………………….

(ii) The probability of an event that cannot happen is …………………………. Such an event is

Called …………………………..

(iii) The probability of an event that is certain to happen is …………………………. Such an event is called ………………………….

(iv) The sum of the probabilities of all the elementary events of an experiment is ………………………….

(v) The probability of an event is greater than or equal to ………………………….and less than or equal to ………………………….

Question 2 : Which of the following experiments have equally likely outcomes? Explain.

(i) A driver attempts to start a car. The car starts or does not start.

(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.

(iii) A trial is made to answer a true-false question. The answer is right or wrong.

(iv) A baby is born. It is a boy or a girl.

Question 3 : Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?

Question 4 : Which of the following cannot be the probability of an event?

(A) 2/3 (B) –1.5 (C) 15% (D) 0.7

Question 5 : If P(E) = 0.05, what is the probability of ‘not E’?

Question 6 : A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out

(i) an orange flavoured candy?

(ii) a lemon flavoured candy?

Question 7 : It is given that in a group of 3 students, the probability of 2 students not having the same  birthday  is  0.992.  What  is  the  probability  that  the  2  students  have  the  same birthday?

Question 8 : A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is (i)red ? (ii) not red?

Question 9 : A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be

(i)                red ?           (ii) white ?       (iii) not green?