Probability Notes for Competitive Exams Preparation

Probability Notes for Competitive Exams Preparation

Probability:

The word Probability is used, in a general sense, to indicate a vague possibility that something might happen. It is also used synonymously with chance.

Random Experiment:

If the result of an experiment conducted any number of times under essentially identical conditions,is not certain but is any one of the several possible outcomes, the experiment is called a trial or a random experiment. Each of the outcomes is know as an event.

Examples:-

I. Drawing 3 cards from a well shuffled pack is a random experiment while getting an Ace or a King are events.

II. Throwing a fair die is a random experiment while getting the score as "2" or an odd number are events.

Mutually Exclusive Events:

If the happening of any one the events in a trial excludes or prevents the happening of all others, then such events are said to be mutually exclusive.

Example:-

The events of getting a head and that of getting a tail when a fair coin is tossed are mutually exclusive.

Equally Likely Event:

Two events are said to be equally likely when chance of occurrence of one eventis equal to that of the other.

Example:-

When a die is throw, any number from 1 to 6 may be got. In this trial, getting any one of these events are equally likely.

Independent Events:

Two events E1 and E2 are said to be independent if the occurrence of the event E2 is not affected by the occurrence or non-occurrence of the event E1.

Example:- 

Two drawings of one ball each time are made from a bag containing balls. Here, we have two events drawing a ball first time (E1) and drawing a ball second time (E2). If the ball of the first draw is replaced in the bag before the second draw is made, then the outcome of E2 does not depend on the outcome of E1. In this case E1 and E2 are Independent events. 

If the ball of the draw is not replaced in the bag before the second draw is made, then the outcome of E2 depends on the outcome of E1. In this case, events E1 and E2 are Dependent events.

Compound Events:

When two or more events are in relation with each other, they are know as Compound Events.

Example:-

When a die is thorow two times, the event of getting 3 in the first throw and 5 in the second throw is a Compound Event.

Definition of Probability:

If an event E can happen in m ways and fail in k ways out of a total of n ways and each of them is equally likely, then the probability of happening E is m/(m+k)=m/n where n=(m+k).

In other words, if a random experiment is conducted n times and m of them are favourable to event E, then the probability of happening of E is P(E)=m/n. since the event does not occur (n-m) times, the probability of non-occurrence of E is P(E).

P(E)=n-m/n= 1-m/n= 1-P(E)

Therefore, P(E)+P(E`)=1.

Note:

1. Probability [P(E)] of the happening of an event E is know as the probability of success and the probability [P(E)] of the non-happening of the event is the probability of failure.
2. If P(E)=1, the event is called a certain event and if P(E)=0 the event is called an impossible event.
3. Instead of saying that the chance of happening of an event is  m/n, we can also say that the odds in favour of the event are m o (n-m) or the odds against the event are (n-m) to m.

Worked Out Examples:

1) When a cubical die is rolled, find the probability of getting an odd integer ?

Sol: There are six possible ways in which the die can fall and of these, three are favourable to the event required. Therefore, required probability= 3/6= 1/2.

2) If a card is draw from a pack, find the probability of getting a king ?

Sol: There are 52 possible ways in which a card can be draw from a pack and of these, four are favourable to the event under discussion. Hence, required probability= 4/52= 1/13.

3) Find the probability of getting at least one 6 when two dice are thrown together ?

Sol: The total number of cases is 6 X 6 = 36. Out of this, we may get 6 on the first die and any of the six numbers on the second die- given six ways. Similarly, 6 on the second die may also be associated with any one of the other five numbers on the first die given 5 ways.

 Total number of occurrences of the event = 11.

Hence required probability = 11/36.

4) When two dice are thrown, find the probability of getting the sum as 10 or 11 ?

Sol: Total number of ways in which two dice can be throw is 36. Out of this, a sum of 10 will come for the combinations of (4, 6), (5, 5) and (6, 4). Sum of  11 will be obtained for (5, 6) and (6, 5).

Hence, total number of ways in which the event can occur is 5. Hence, required probability is 5/36.