**Dear Readers,**

Today we’ll discuss about Probability. This topic is can fetch you marks easily but you need to know the right concepts and types of questions to practice.

**All about Probability**

**Probability**

Probability theory deals in random events. The central objects of probability theory are random variables, and events.

Event is generally defined as outcome of an experiment. Events can be classified as deterministic or probabilistic (random).

**Deterministic Event**

When an experiment is repeated under homogeneous conditions and it produces the same results, then the experiment is known as deterministic experiment.

**☞****For example** if a car runs at a speed of 50 km/hr under uninterrupted condition it will reach a place 100 km distant in 2 hr.

**Random or Probabilistic Event**

An experiment when repeated under identical condition does not produce the same result every time but out come is one of the several possible out comes, then such an experiment is known as a probabilistic experiment or a random experiment and these events are known as random events.

**☞****For example** if it is raining today. It may not rain tomorrow. If a coin is tossed, outcome may be a head or tail.

**Sample Space**

The sample space or universal sample space, often denoted S or U (for “universe”), of an experiment or random trial is the set of all possible outcomes.

**Independent Events**

If one event does not have any effect on other, then these events are known as independent events. If A and B are independent events then,

**Example: A man throws two identical unbiased die. If he gets 3 on the first dice or an even number on the second, he wins. Find the probability that the man wins.**

**Solution: **A = Event of getting 3 on the first dice and B = Event of getting an even number on the second dice

**Conditional Probability**

A conditional probability is the probability of an event given that another event has occurred. Suppose a card is randomly drawn from the deck of 52 cards and it is found a red card, the probability of finding that this card is a KING, is an example of conditional probability.

If event E2 has occurred already then probability of occurrence of E1 is denoted by

**Example:** **A number is selected randomly from the set of numbers ranging from 1 to 100 number is found to be a multiple of 3. Find the probability that it’s a multiple of 7 also.**

**Solution:** Method 1: Sample space S = {1, 2, 3………..100}

Given that the selected number is a multiple of 3, hence conditional sample space S_1 = {3, 6,………99}.

Now the number should be a multiple of 3 as well as 7, hence it must be a multiple of 21. There are 4 such numbers: 21, 42, 63 and 84.

Required sample space S_2 = {21, 42, 63, 84}

Probability = S2/S1 =4/33

Method 2: Suppose E2 is the event that number is a multiple of 3 and E1 is the event that number is a multiple of 7, then

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