Solution of Chapter 13. Statistics and Probability (NCERT - Mathematics Exemplar Book)

In the formula for finding the mean of grouped data d_i’s are deviation from a of

While computing mean of grouped data, we assume that the frequencies are

If x_i’s are the mid - points of the class intervals of grouped data, f_i are the corresponding frequencies and is the mean, then is equal to

In the formula for finding the mean of grouped frequency distribution is equal to

The abscissa of the point of intersection of the less than type and of the more than type cumulative frequency curves of a grouped data gives its

For the following distribution,

the sum of lower limits of the median class and modal class is

Consider the following frequency distribution

The upper limit of the median class is

For the following distribution,

the modal class is

Consider the data

The difference of the upper limit of the median class and the lower limit of the modal class is

The times (in seconds) taken by 150 athletes to run a 110m hurdle race are tabulated below

The number of athletes who completed the race in less than 14.6 s is

Consider the following distribution

The frequency of the class 30 - 40 is

If an event cannot occur, then its probability is

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

An event is very unlikely to happen. Its probability is closest to

If the probability of an event is P, then the probability of its complementary event will be

The probability expressed as a percentage of a particular occurrence can never be

If P(A) denotes the probability of an event A then

If a card is selected from a deck of 52 cards, then the probability of its being a red face card is

The probability that a non - leap your selected at random will contains 53 Sunday is

When a die is thrown, the probability of getting an odd number less than 3 is

A card is drawn from a deck of 52 cards. The event E is that card is not an ace of hearts. The number of outcomes favorable to E is

The probability of getting a bad egg in a lot of 400 is 0.035. The number of bad eggs in the lot is

A girl calculates that the probability of her winning the first prize in a lottery is 0.08. If 6000 tickets are sold, then how many tickets has the bought?

One ticket is drawn at random from a bag containing tickets numbered 1 to 40. The probability that the selected ticket has a number which is a multiple of 5 is

Someone is asked to take a number from 1 to 100. The probability that it is a prime, is

A school has five houses A,B,C,D and E. A class has 23 students, 4 from house A, 8 from house B, 5 from house C, 2 from house D and rest from house E. A single student is selected at random to be the class monitor. The probability that the selected student is not from A, B and C is

The median of an ungrouped data and the median calculated when the same data is grouped are always the same. Do you think that this is a correct statement? Give reason.

In calculating the mean of grouped data, grouped in classes of equal width, we may use the formula,

Where, is the assumed mean, must be one of the mid - point of the classes. Is the last statement correct? Justify your answer.

Is it true to say that the mean, mode and median of grouped data will always be different? Justify your answer.

Will the median class and modal class of grouped data always be different? Justify your answer.

In a family having three children, there may be no girl, one girl, two girls or three girls. So, the probability of each is 1/4. Is this correct? Justify your answer.

A game consists of spinning an arrow which comes to rest pointing at one of the regions (1, 2 or 3) (see figure). Are the outcomes 1, 2, and 3 equally likely to occur? Give reasons.

Apoorv throws two dice once and computes the product of the numbers appearing on the dice. Peehu throws one die and squares the number that appears on it. Who has the better chance of getting the number 36? Why?

When we toss a coin, there are two possible outcomes - head or tail. Therefore, the probability of each outcome is . Justify your answer.

A student says that, if you throw a die, it will show up 1 or not 1. Therefore, the probability of getting 1 and the probability of getting not 1 each is equal to Is this correct? Give reasons.

I toss three coins together. The possible outcomes are no heads, 1 head, 2 head and 3 heads. So, I say that probability of no heads is What is wrong with this conclusion?

If you toss a coin 6 times and it comes down heads on each occasion. Can you say that the probability of getting a head is 1? Given reasons.

Sushma tosses a coin 3 times and gets tail each time. Do you think that the outcome of next toss will be a tail? Give reason.

If I toss a coin 3 times and get head each time, should I expect a tail to have a higher change in the 4^th toss? Given reason in support of your answer.

A bag contains slips numbered from1 to 100. If Fatima chooses a slip at random from the bag, it will either be an odd number or an even number. Since, this situation has only two possible outcomes, so the probability of each is 1/2.Justify.

Find the mean of the distribution

Calculate the mean of the scores of 20 students in a mathematics test

Calculate the mean of the following data

The following table gives the number of pages written by Sarika for completing her own book for 30 days.

Find the mean number of pages written per day.

The daily income of a sample of 50 employees are tabulated as follows.

Find the mean daily income of employees.

An aircraft has 120 passenger seats. The number of seats occupied during 100 flights is given in the following table.

Determine the mean number of seats occupied over the flights.

The weights (in kg) of 50 wrestlers are recorded in the following table.

Find the mean weight of the wrestlers.

The mileage (km per liter) of 50 cars of the same model was tested by a manufacturer and details are tabulated as given below

Find the mean mileage.

The manufacturer claimed that the mileage of the model was 16Km L^-1

Do you agree with this claim?

The following is the distribution of weights (in kg) of 40 persons.

Construct a cumulative frequency distribution (of the less than type) table for the data above.

The following table shows the cumulative frequency distribution of marks of 800 students in an examination.

Construct a frequency distribution table for the data above.

From the frequency distribution table from the following data

Find the unknown entries a,b,c,d,e and f in the following distribution of heights of students in a class

The following are the ages of 300 patients getting medical treatment in a hospital on a particular day

Form

(i) Less than type cumulative frequency distribution.

(ii) More than type cumulative frequency distribution.

Given below is a cumulative frequency distribution showing the marks secured by 50 students of a class

Form the frequency distribution table for the data.

Weekly income of 600 families is tabulated below

Compute the median income.

The maximum bowling speeds, in km per hour, of 33 players at a cricket coaching center are given as follows

Calculate the median bowling speed.

The monthly income of 100 families are given as below

Calculate the modal income.

The weight of coffee in 70 packets are shown in the following table

Determine the model weight.

Two dice are thrown at the same time. Find the probability of getting

(i) same number on both dice.

(ii) different number on both dice.

Two dice are thrown simultaneously. What is the probability that the sum of the numbers appearing on the dice is

(i) 7?

(ii) a prime number?

(iii) 1?

Two dice are thrown together. Find the probability that the product of the numbers on the top of the dice is

(i) 6

(ii) 12

(iii) 7

Two dice are thrown at the same time and the product of numbers appearing on them is noted. Find the probability that the product is less than 9.

Two dice are numbered 1, 2, 3, 4, 5, 6 and 1, 1, 2, 2, 3, 3, respectively. They are thrown and the sum of the numbers on them is noted. Find the probability of getting each sum from 2 to 9, separately.

A coin is tossed two times. Find the probability of getting at most one head.

A coin is tossed 3 times. List the possible outcomes. Find the probability of getting

(i) all heads

(ii) at least 2 heads

Two dice are thrown at the same time. Determine the probability that the difference of the numbers on the two dice is 2.

A bag contains 10 red, 5 blue and 7 green balls. A ball is drawn at random. Find the probability of this ball being a

(i) red ball

(ii) green ball

(iii) not a blue ball

The king, queen and jack of clubs are removed from a deck of 52 playing cards and then well shuffled. Now, one card is drawn at random from the remaining cards. Determine the probability that the card is

(i) a heart

(ii) a king

Refer to 0.28. What is the probability that the card is

(i) a club

(ii) 10 of hearts

All the jacks, queens and kings are removed from a deck of 52 playing cards. The remaining cards are well shuffled and then one card is drawn at random. Giving ace a value 1 similar value for other cards, find the probability that the card has a value.

(i) 7

(ii) greater than 7

(iii) less than 7

An integer is chosen between 0 and 1000. What is the probability that it is

(i) divisible by 7?

(ii) not divisible by 7?

Cards with numbers 2 to 101 are placed in a box. A card is selected at random. Find the probability that the card has

(i) an even number

(ii) a square number

A letter of English alphabets is chosen at random. Determine the probability that the letter is a consonant

There are 1000 sealed envelopes in a box, 10 of them contain a cash prize of Rs 100 each, 100 of them contain a cash prize of `Rs50 each and 200 of them contain a cash prize of `Rs10 each and rest do not contain any cash prize. If they are well shuffled and an envelope is picked up out, what is the probability that it contains no cash prize?

Box A contains 25 slips of which 19 are marked Re 1 and other are marked Rs 5 each. Box B contains 50 slips of which 45 are marked Re 1 each and others are marked Rs13 each. Slips of both boxes are poured into a third box and reshuffled. A slip is drawn at random. What is the probability that it is marked other than Re1?

A carton of 24 bulbs contain 6 defective bulbs. One bulb is drawn at random. What is the probability that the bulb is not defective? If the bulb selected is defective and it is not replaced and a second bulb is selected at random from the rest, what is the probability that the second bulb is defective?

A child’s game has 8 triangles of which 3 are blue and rest are red, and 10 squares of which 6 are blue and rest are red. One piece is lost at random. Find the probability that it is a

(i) triangle

(ii) square

(iii) square of blue colour

(iv) triangle of red colour

In a game, the entry fee is of Rs.5. The game consists of a tossing a coin 3 times. If one or two heads show, Sweta gets her entry fee back. If she throws 3 heads, she receives double the entry fees. Otherwise she will lose. For tossing a coin three times, find the probability that she

(i) loses the entry fee.

(ii) gets double entry fee.

(iii) just gets her entry fee.

A die has its six faces marked 0, 1, 1, 1, 6, 6. Two such dice are thrown together and the total score is recorded.

(i) How many different scores are possible?

(ii) What is the probability of getting a total of 7?

A lot consists of 48 mobile phones of which 42 are good, 3 have only minor defects and 3 have major defects. Varnika will buy a phone, if it is good but the trader will only buy a mobile, if it has no major defect. One phone is selected at random from the lot. What is the probability that it is

(i) acceptable to Varnika?

(ii) acceptable to the trader?

A bag contains 24 balls of which x are red, 2x are white and 3x are blue. A ball is selected at random. What is the probability that it is

(i) not red?

(ii) white

At a fete, cards bearing numbers 1 to 1000, one number on one card, are put in a box. Each player selected one card at random and that card is not replaced. If the selected card has a perfect square greater than 500, the player wins a prize. What is the probability that

(i) the first player wins a prize?

(ii) the second player wins a prize, if the first has won?

Find the mean marks of students for the following distribution

Determine the mean of the following distribution

Find the mean age of 100 residents of a town from the following data.

The weights of tea in 70 packets are shown in the following table

Find the mean weight of packets.

Refer to Q. 4 above. Draw the less than type ogive for this data and use it to find the median weight.

Refer to Q.5 above. Draw the less than type and more than type ogives for the data and use them to find the median weight.

The table below shows the salaries of 280 persons.

Calculate the median and mode of the data.

The mean of the following frequency distribution is 50 but the frequencies f₁ and f₂ in classes 20 - 40 and 60 - 80, respectively are not known. Find these frequencies, if the sum of all the frequencies is 120.

The median of the following data is 50. Find the values of p and q, if the sum of all the frequencies is 90.

The distribution of heights (in cm) of 96 children is given below

Draw a less than type cumulative frequency curve for this data and use it to compute median height of the children.

Size of agricultural holdings in a survey of 200 families is given in the following table

Compute median and mode size of the holdings.

The annual rainfall record of a city for 66 days is given in the following table.

Calculate the median rainfall using ogives (or move than type and of less than type)

The following is the frequency distribution of duration for 100 calls made on a mobile phone.

Calculate the average duration (in sec) of a call and also find the median from a cumulative frequency curve.

50 students enter for a school javelin throw competition. The distance (in metre) thrown are recorded below

(i) Construct a cumulative frequency table.

(ii) Draw a cumulative frequency curve (less than type) and calculate the median distance drawn by using this curve.

(iii) Calculate the median distance by using the formula for median.

(iv) Are the median distance calculated in (ii) and (iii) same?