Probabilty

Understand the terms related to probability

Suppose there are two pens of different colors in a box. One is blue and the other is black. **If you draw a pen from the box, without looking at the colour, then can you say that it will be the blue pen or the black pen surely?**

You can never say with surety whether the pen will be of blue color or black color. Thus, there are two possible outcomes and there is an equal chance of both the possible outcomes to occur.

Since the two possible outcomes have equal chances of occurrence, the chance for each outcome to occur is half. Thus, the probability of drawing a blue pen or a black pen from the box will be same and equal to.

Now, consider the case that the box contains one blue pen and two black pens.

Here, the probability of drawing a black pen will be more than the probability of drawing a blue pen. This is because the number of black pens is more than the number of blue pens in the box.

Further, the number of black pens is twice the number of blue pens. Thus, the probability of drawing a black pen is twice the probability of drawing a blue pen.

The probability of drawing a blue pen is and the probability of drawing a black pen is.

Let us consider one more example to understand the concept better.

Sanjana throws a dice. She can get any of the numbers: 1, 2, 3, 4, 5, or 6 on the top face of the dice. There is no other possibility. Moreover, there is an equal chance of getting these numbers, i.e., the probability of getting any of these numbers is the same. Since the number of possible outcomes is 6, the probability of getting any of these numbers is.

Now, when she throws a dice, can she get the number 7 on the top face of the dice?

Observe that when a dice is thrown, then we cannot get the number 7. Thus, the probability of getting the number 7 is nothing but zero.

Thus, **the probability of the event which has no possibility to occur is 0.**

Also, observe that the probability of getting any of the numbers 1, 2, 3, 4, 5, and 6 on the top face of the dice is 1 as we will surely get any one of the numbers from 1 to 6 upon throwing a die.

Thus, we can say that **the probability of the event which is sure to occur is always 1.**

**From the above examples, we conclude the following facts:**

**1)** **The probability of occurrence of any event always lies between 0 and 1.**

**2) ****The probability of such an event which has no possibility to occur is 0**.

**3) ****The probability of such an event which is sure to occur is 1.**

Let us look at some more examples now.

**Example 1:**

**When a coin is tossed, what is the probability of getting a head?**

**Solution:**

When a coin is tossed, there are two possible outcomes − head or tail. The possibility of occurrence of both of them will be the same. Therefore, the probability of getting a head is equal to the probability of getting a tail. Thus, the probability of getting a head is equal to.

**Example 2:**

**A dice is thrown. What is the probability of getting**

**the number 2****any one of the number among 1, 2, 3, 4, 5, and 6****the number 7**

**Solution:**

- When a dice is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, and 6 and all of them will have the same possibility of occurrence as any of them can come when a dice is thrown.

Thus, the probability of getting the number 2 is.

- When a dice is thrown, we can get any one of the numbers 1, 2, 3, 4, 5, and 6.

Thus, the probability of getting any one of the numbers among 1, 2, 3, 4, 5, and 6 is 1.

- When a dice is thrown, there is no possibility of getting the number 7, as the number 7 is not there on the dice.

Thus, the probability of getting the number 7, when a dice is thrown, is 0.

**Example 3:**

**Which of the following cannot be the probability of an event?**

**(i) **

**(ii) −0.5**

**(iii) **

**Solution:**

We know that the probability of an event *E* always lies between 0 and 1.

0 ≤ *P*(*E*) ≤ 1

**(i) **= 0.11

0 ≤ 0.11≤ 1

Thus, can be the probability of an event.

**(ii) **−0.5 does not lie in the range of 0 ≤ *P*(*E*) ≤ 1.

Thus, it cannot be the probability of an event.

**(iii)** = 1.18, which does not lie in the range of 0 ≤ *P*(*E*) ≤ 1.

Thus, it cannot be the probability of an event.

Consider the experiment of throwing a dice. Any of the numbers 1, 2, 3, 4, 5, or 6 can come up on the upper face of the dice. We can easily find the probability of getting a number 5 on the upper face of the dice?

Mathematically, probability of any event *E* can be defined as follows.

Here, *S* represents the sample space and *n*(*S*) represents the number of outcomes in the sample space.

For this experiment, we have

Sample space (*S*) = {1, 2, 3, 4, 5, 6}. Thus, *S* is a finite set.

So, we can say that the possible outcomes of this experiment are 1, 2, 3, 4, 5, and 6.

Number of all possible outcomes = 6

Number of favourable outcomes of getting the numbe…

To view the complete topic, please