Probability

Probability in everyday life, we come across statements such as:

Most probably it will rain today.
Chances are high that the prices of petrol will go up.
I doubt that he will win the race.

The words ‘most probably’, ‘chances’, ‘doubt’ etc., show the probability of occurrence of an event.

Some Terms Related to Probability

Experiment:

An operation which can produce some well-defined outcomes is called an experiment. Each outcome is called an event.

Random Experiment:

In an experiment where all possible outcomes are known and in advance if the exact outcome cannot be predicted, is called a random experiment.

Thus, when we throw a coin we know that all possible outcomes are Head and Tail.
But, if we throw a coin at random, we cannot predict in advance whether its upper face will show a head or a tail.

So, tossing a coin is a random experiment.
Similarly, throwing a dice is a random experiment.

To know more about random experiments in details Click Here.

Trial:

By a trial, we mean performing a random experiment.

For example; throwing a die or tossing a coin etc.

Sample space:

A sample space of an experiment is the set of all possible results of that random experiment.

For example; in throwing a die possible results are {1, 2, 3, 4, 5, 6}.

Event:

Out of the total results obtained from a certain experiment, the set of those results which are in favor of a definite result is called the event and it is denoted as E.

Equally Likely Events:

When there is no reason to expect the happening of one event in preference to the other, then the events are known equally likely events.

For example; when an unbiased coin is tossed the chances of getting a head or a tail are the same.

Exhaustive Events:

All the possible outcomes of the experiments are known as exhaustive events.

For example; in throwing a die there are 6 exhaustive events in a trial.

Favorable Events:

The outcomes which make necessary the happening of an event in a trial are called favorable events.

For example; if two dice are thrown, the number of favorable events of getting a sum 5 is four,

i.e., (1, 4), (2, 3), (3, 2) and (4, 1).

Additive Law of Probability:

If E₁ and E₂ be any two events (not necessarily mutually exclusive events), then P(E₁ ∪ E₂) = P(E₁) + P(E₂) - P(E₁ ∩ E₂)

Probability of Occurrence of an Event:

The probability of occurrence of an event is defined as:

P(occurrence of an event)

Number of trials in which event occurred
⁼ Total number of trials

Solved examples on Probability:

1. A dice is thrown 65 times and 4 appeared 2 1 times. Now, in a random throw of a dice, what is the probability of getting a 4?

Solution:

Total number of tria1s = 65.

Number of times 4 appeared = 21.

Probability of getting a 4 = ^{Number of times 4 appeared}/_{Total number of trials}

= ²¹/₆₅

2. A survey of 200families shows the results given below:

No. of girls in the family	2	1	0
No. of Families	32	154	14

Out of these families, one is chosen at random. What is the probability that the chosen family has 1 girl?

Solution:

Total number of families = 200.

Number of families having 1 girl = 154.

Probability of getting a family having 1 girl

   = ^{Number of families having 1 girl}/_{Total number of families}

   = ¹⁵⁴/₂₀₀

   = ⁷⁷/₁₀₀

Worksheet Probability:

1. The tree diagram above represents three events. In the first event either a Red, White, or Blue circle is chosen. In the second event either a Red, White, or Blue circle is chosen. In the third event either a Red, White, or Blue circle is chosen.

Match the following events with the corresponding probabilities:

(a) The second circle is white (a) 10/15

(b) All three circles are red (b) 4/15

(d) At least two circles are the same (d) 3/15

(e) The first circle is not red (e) 1/15

(f) The first two circles are blue (f) 12/15

(g) The third circle is blue (g) 15/15

2. The tree diagram above represents three events. In the first event either an A, B, or C is chosen. In the second event either an A, B, or C is chosen. In the third event either a D, E, or F is chosen.

Match the outcome with its probability:

(a) The second letter is a C (a) 6/12

(b) The first or second letter is an A (b) 0/12

(d) The first two letters chosen are both A (d) 3/15

(e) All three letters are the same (e) 1/15

(f) The first letter is not an A (f) 12/15

(g) ADD (g) 15/15

Probability

Random Experiments

Experimental Probability

Events in Probability

Empirical Probability