**NCERT Class 9 Maths Lab Manual – Find Probability of Unit’s Digit of Telephone Numbers**

**Objective**

To find experimental probability of unit’s digits of telephone numbers listed on a page selected at random of a telephone directory.

**Materials Required**

- Telephone directory
- Ruler
- Notebook
- Pen

**Prerequisite Knowledge**

Basic knowledge of probability.

**Theory**

- If E is an event that happen when an experiment is performed, then the experimental or empirical probability of the event E is given by

P(E) = \(\frac { Number\quad oftrials\quad in\quad which\quad the\quad event\quad E\quad happened }{ Total\quad number\quad of\quad trials } =\frac { n(E) }{ n(S) }\)

or

Probability of an event E = P(E) = \(\frac { Number\quad of\quad trials\quad in\quad which\quad the\quad event\quad occurred }{ Total\quad number\quad of\quad trials }\) - The probability of happening of an event always lies from 0 to 1, i.e. 0 ≤ P(E) ≤ 1.

In percentage, it lies from 0% to 100%. - If probability of an event say A is 1, i.e. P(A) = 1, then event A is called a certain event or sure event.
- If probability of an event say B is 0, i.e. P(B) = 0, then event B is called an impossible event.
- The sum of all the probabilities of all possible outcomes of an experiment is 1.

**Procedure**

- Taking a telephone directory, select any page at random.
- Suppose the count of total telephone numbers on the selected page is N.
- Unit place of any telephone number can be occupied by any one of the digits 0,1,2, ……. 9.
- Now, using tally marks, prepare a frequency distribution table for the digits at unit’s place.
- Now, using the table, write the frequency of each of the digits 0,1,2, …….. 9.
- By using the formula for experimental probability, find the probability of each digit.

**Demonstration**

1. Firstly, by using tally marks, prepare a frequency distribution table for the digits 0,1,2,…. 9

Digits |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Tally marks frequency |
n_{0} |
n_{1} |
n_{2} |
n_{3} |
n_{4} |
n_{5} |
n_{6} |
n_{7} |
n_{8} |
n_{9} |

2. From the table, note down the frequency of each digit from 0 to 9.

3. We get that digits 0, 1,2,…, 9 are occurring n0, n1, n2, …, n9 times respectively.

4. Considering the occurrence of each digit as an event E, the probability of event E is

P(E) = \(\frac { Number\quad of\quad trials\quad in\quad which\quad event\quad occurred }{ Total\quad number\quad of\quad trials }\)

Hence, respective experimental probability of occurrence of 0, 1, 2, …, 9 is given by

P(0) = \(\frac { { n }_{ 0 } }{ N }\), P(1) = \(\frac { { n }_{ 1 } }{ N }\), ……., P(9) = \(\frac { { n }_{ 9 } }{ N }\)

**Observations**

Total telephone numbers on a page (N) = ………….

Number of times 0 occurring at unit’s place (n_{0}) = …………..

Number of times 1 occurring at unit’s place (n_{1}) = …………….

Number of times 2 occurring at unit’s place (n_{2}) = …………..

Number of times 3 occurring at unit’s place (n_{3}) = ………….

Number of times 4 occurring at unit’s place (n_{4}) = ………….

……………..

……………….

Number of times 9 occurring at unit’s place (n_{9}) = …………….

Hence, experimental probability of occurrence of 0 = P(0) = \(\frac { { n }_{ 0 } }{ N }\)

Now, experimental probability of occurrence of 1 = P(1) = \(\frac { { n }_{ 1 } }{ N }\)

P(2) = \(\frac { { n }_{ 2 } }{ N }\)

……….

……….

P(9) = \(\frac { { n }_{ 9 } }{ N }\)

**Result**

We have got the experimental probability of unit’s digits of telephone numbers listed on a page selected at random of a telephone directory.

**Applications**

The concept of experimental probability is useful in

- deciding premium tables by insurance companies. .
- stock market to forecast the performance of a company, by metreological department to forecast weather.

**Viva-Voce**

**Question 1.**

How will you define an event?

**Answer:**

An event for an experiment is the collection of some outcomes of the experiment.

**Question 2.**

How will you define the empirical probability P(E) of an event E?

**Answer:**

P(E) = \(\frac { Number\quad of\quad trials\quad in\quad which\quad E\quad has\quad happened }{ Total\quad number\quad of\quad trials }\)

**Question 3.**

What are the maximum and minimum values of the probability of an event?

**Answer:**

Maximum and minimum values of the probability of an event are 1 and 0 respectively.

**Question 4.**

What is the complement of an event E?

**Answer:**

1 – P(E)

**Question 5.**

What is the probability of a certain event?

**Answer:**

1

**Question 6.**

How many events can occur when a coin is tossed?

**Answer:**

Two events, i.e. head or tail.

**Question 7.**

How will you define a sure event?

**Answer:**

If probability of an event say A is 1, i.e. P(A) = 1, then event A is called a certain event or a sure event.

**Question 8.**

Is the sum of all the probabilities of all possible outcomes of an experiment 1?

**Answer:**

Yes, the sum of all the probabilities of all possible outcomes of an experiment is 1.

**Suggested Activity**

Find the experimental probability of getting a tail in tossing an unbiased coin 5,10,15,20,25,30 times.

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