Class 9 Data Handling - Probability, Statistics

Transcription

1 ID : in-9-data-handling-probability-statistics [1] Class 9 Data Handling - Probability, Statistics For more such worksheets visit Answer t he quest ions (1) A card is drawn f rom a shuf f led deck of 5 cards. What is the probability of getting a card greater than 3 but lesser than 8? () What is the average of the 7 consecutive integers starting at 17? (3) Find the mean of all the internal angles in a triangle. (4) The numbers 1 to 9 are written on 9 pieces of paper and dropped into a box. Three of them are drawn at random. What is the probability that the three pieces of paper picked have numbers that are in arithmetic progression? (5) You can f orm exactly 6 three digit numbers using the digits 8, 6 and 7 exactly once in one number. What is the average of these six numbers? (6) In 008, the meteorological of f ice predicted the weathers completely right f or the months of f ebruary and april, and completely wrong f or all the other months. What is the probability that the f orecast was wrong f or a given day that year? (7) A teacher measures the heights of 13 students in her class as f ollows (in centimetres) 157, 158, 119, 146, 14, 118, 117, 159, 117, 143, 10, 150, 154 What is the median of their heights? (8) Shilpa selects 3 numbers randomly f rom the f ollowing set of 5 numbers 4, 3,, 1 and 5. He puts them in the f orm of a proper f raction of the type a b c. What is the probability that you will get a f raction greater than 15 16? Choose correct answer(s) f rom given choice (9) A box contains 6 orange and 3 purple balls. Vinayak takes out a ball, puts it back, and then takes out another ball. What is the probability that both balls were orange? a b c. 6 9 d

2 (10) From a deck of cards, Ankur took out one card at random. What is the probability that he got a prime number? ID : in-9-data-handling-probability-statistics [] a. 13 b c d (11) While doing the science experiment in the physics lab, Raj had to take 7 measurements of the temperature and write the average of those as an answer. If the measurements of the temperature is -1, -3, 0, -4, -, -3,, what is the f inal answer of his experiment? a. -11 b. -77 c d (1) Ridhima got an average score of 77.5 in 4 tests. She got 78 as the average of the highest 3 scores, and his lowest two scores are the same numbers. What is the average of her highest two scores? a. 78 b. 79 c d (13) Find the median of the f ollowing set 15, 1, 0, 0, 1, 17, 10, 10, 15, 16 a b c d Fill in the blanks (14) A ship docks at the port on 5 dif f erent days at 11:15 AM, 11:00 AM, 10:00 AM, 4:45 AM and 5:45 AM. The average time it docks is : AM (15) The average of Sneha's marks in 4 subjects is 89. She got 34 marks in 5 th subject. Her average in all 5 subjects is. 016 Edugain ( All Rights Reserved Many more such worksheets can be generated at

3 Answers ID : in-9-data-handling-probability-statistics [3] (1) 16 5 To f igure this out, we identif y how many cards that match this condition are present in the deck There are f our suits, and each suit has 13 cards : 1 to 9, J, Q, K and A The cards that are > 3 but < 8 in each suit are 4,...,7 The number of such cards is 4 Step 4 Since there are 4 such suits, the total probability is 4 x 4 5 = 16 5 () 0 If you look at the question caref ully, you will notice that the 7 consecutive integers starting at 17 = 17, 18, 19, 0, 1,, 3 Sum of 7 consecutive integers = = 140 Average of 7 consecutive integers starting at 17 = Sum of 7 consecutive integers T otal number of consecutive integers = = 0 Now the average of 7 consecutive integers at 17 is 0. (3) 60 The sum of all three internal angles in a triangle must be 180. Now the mean of all the internal angles in a triangle = 180 = 60 3 Theref ore the mean of all the internal angles in a triangle is 60

4 (4) 1 ID : in-9-data-handling-probability-statistics [4] 63 This is a little complicated, so f ollow caref ully For making the explanation and the equations simpler, think of the number on the pieces of paper in the f orm of (n+1) Here, we can see f rom the equation n+1 = 9, so n=4 The probability of getting 3 numbers in an A.P by selecting 3 numbers randomly between 1 and 9 is the ratio of - Number of ways we can get an A.P f rom 3 random numbers between 1 to 9, and - Number of ways to select 3 random numbers between 1 to 9 Step 4 Let's look at the second part f irst. Three tickets can be drawn f rom (n+1) numbers is in [( x n) + 1] C 3 ways i.e. Number of ways 3 tickets can be drawn = (n+1)(n)(n-1) 3xx1 Simplif ying this, we get the number of ways to draw 3 numbers between 1 and 9 = n(4n -1) 3 Here n = 4, so we can simplif y it as 84 Step 5 Now f or the ways we can get an A.P f rom 3 numbers bwetween Arithmetic Progressions of 3 numbers would be a sequence of 3 numbers that are separated by a common interval e.g. 1,,3 or 3,5,7 etc. They are in the f orm (a, a+d, a + d), where a is an integer f rom 1 to (9-), and d is another integer So it's helpf ul to think of the solution in terms of this interval. So we'll think of all the sequences that have an interval 1, then sequences with interval, and so on Step 6 So what are the possible sequences with interval 1. They are (1,,3) (,3,4)... (n-1,n,n+1) T here are theref ore n-1 such possible sequences Step 7 Similarly, let's look at A.P with interval between the terms. They are (1,3,5) (,4,6)

5 ...<n-3,n-1,n+1> T here are n-3 such possible sequences ID : in-9-data-handling-probability-statistics [5] Step 8 We can generalize this to say that the number of such sequences with interval 'd' is (n- (d-1)) Obviously the largest possible integer is d=n, with just one sequence (1, n+1, n+1) Step 9 So the total number of such sequences is (n-1) + (n-3) + (n-5) This is itself an AP with n terms and d= The sum of this sequence is n [ + (n-1)] Simplif ying, we get n Here n=4, so this is 16 0 So the probability is 1 63 (5) 777 (6) The key thing to note here is that 008 is a leap year A leap year has 366 days Now we need to f igure out the number of days in f ebruary and april f ebruary has 9 days and april has 30 days Adding them together we get = 59 days Step 4 So the f orecast was right f or 59 days and wrong f or 307 days Step 5 The probability that the f orecast was wrong on a given day would theref ore be

6 (7) 143 ID : in-9-data-handling-probability-statistics [6] If you look at the question caref ully, you will notice that a teacher measures the heights of 13 students in her class as f ollows (in centimetres) 157, 158, 119, 146, 14, 118, 117, 159, 117, 143, 10, 150, 154 Median is the middle number in a sorted list. To f ind the median f irst of all arrange the heights in the ascending order, we get 117, 117, 118, 119, 10, 14, 143, 146, 150, 154, 157, 158, 159 (n + 1) Median = ( ) th [if n is odd] (where n is the number of terms ) ( ) th = ( 14 ) th = 7 th Since 7 th term in the scores 117, 117, 118, 119, 10, 14, 143, 146, 150, 154, 157, 158, 159 is 143, theref ore median = 143 Theref ore the median of heights of 13 students is 143.

7 (8) 18 ID : in-9-data-handling-probability-statistics [7] 30 We need to select 3 numbers out of the 5 given in order to get a f raction of the f orm a b c We are also told it is a proper f raction, so b should be greater than c Let's put the integers in a sorted manner. We get 1,, 3, 4 and 5 Now we need to see how many proper f ractions can be f ormed f rom them b A proper f raction of the type a has three integers, a whole number a, a numerator b, c and a denominator c Let's assume we use one of the integers in the list above as the whole number a We now can select b and c f rom the remaining 4 integers We can select integers f rom 4 in 4C = 4x3 = 6 ways x1 For any pair we select, one will be greater than the other, and the smaller integer will f orm the numerator and the larger one the denominator - Note: that the other way won't work - if the numerator is larger than the denominator it is not a proper f raction So f or each of the 5 integers, if we select one as the whole number, we get 6 possible combinations of numerator and denominator that can f orm a proper f raction This means there are 5x6 = 30 possible proper f ractions of the f orm a b c that can be f ormed f rom these 5 integers Step 4 Now we need to f igure out how many of these 30 f ractions are greater than In gt, we see that the whole number is, the numerator is 15 and the denominator is 16 We see that the numerator 15 is larger than the largest number in the set of numbers given to us, and the denominator 16 is one larger than The implication of this is that will be larger than any f raction that can be f ormed 16 f rom the set of numbers 1,, 3, 4 and 5 where the whole number of the f raction is So we only need to count the f ractions that have the whole number greater than These are the f ractions that will have the whole numbers as 3, 4 and 5 Step 5 Now, remember there are 30 proper f ractions you can f orm f rom this list of number From our analysis above, we also saw that f or each number selected as the whole number,

8 we can f orm 6 f ractions f rom this list of numbers ID : in-9-data-handling-probability-statistics [8] Step 6 So using each of the numbers f rom 3, 4 and 5 as the whole number, we can f orm 6 proper f ractions The total number of f ractions that can be f ormed using 3, 4 and 5 as the whole number = 6 x 3 = 18 Step 7 Out of the 30 f ractions, 18 will be greater than Step 8 The probability is theref ore = 18 30

9 (10) d ID : in-9-data-handling-probability-statistics [9] There are 5 cards in a deck, with 13 of each suits (diamonds, clubs, hearts and spades Each of the 13 cards of a suits are the numbers 1 to 9, then 4 f ace cards, including the ace In the numbers 1 to 9, only,3,5,7 are primes. So f our primes per suit, which gives a total of 16 prime cards Step 4 So there are 16 possible prime cards out of the total 5. The probability that we pick a prime card is theref ore 16 4, which can be reduced to 5 13 (11) d If you look at the question caref ully, you will notice that Raj had to take 7 measurements of the temperature and write the average of those as an answer. Since the measurements of the temperature is -1, -3, 0, -4, -, -3,, theref ore the f inal answer of his experiment = average of -1, -3, 0, -4, -, -3,, Sum of -1, -3, 0, -4, -, -3,, = Number of measurements of the temperature = = (1) b. 79

10 (13) d ID : in-9-data-handling-probability-statistics [10] If you look at the question caref ully, you will notice that the given data is 15, 1, 0, 0, 1, 17, 10, 10, 15, 16 To f ind the median f irst of all arrange the data in the ascending order, you get 10, 10, 1, 15, 15, 16, 17, 0, 0, 1 Total number of terms are 10 which is even. So, median is equal to the average of ( n ) th and ( n +1) th terms, (where n is the number of terms) ( n ) th = ( 10 ) th = 5 th ( n +1) th = ( ) th = 6 th 5 th and 6 th terms are 15 and 16 respectively median = = 31 = 15.5 Step 4 Theref ore the median of the data set is 15.5.

11 (14) 8 33 ID : in-9-data-handling-probability-statistics [11] Since all the times are in AM, we can just add the hours in minutes f orm and f ind out the average We take a time as the minutes past midnight So 11:15 AM = 11 x = :00 AM = 11 x = :00 AM = 10 x = 600 4:45 AM = 4 x = 85 5:45 AM = 5 x = 345 To f ind the average = = 513 To convert it back to the hh:mm f ormat we divide by 60. The quotient is the hours, and the remainder is the minutes 513 = 8R33 60 Average time of docking = 8:33 AM

12 (15) 78 ID : in-9-data-handling-probability-statistics [1] We know that the average of marks in all subjects = Sum of marks in all subjects T otal number of subjects. The average of Sneha's marks in 4 subjects = Sum of marks in 4 subjects = 4 89 = 356. Sum of marks in 4 subjects 4 = 89 Since she got 34 marks in the 5 th subject, the average of Sneha's marks in 5 subjects = Sum of marks in 5 subjects 5 = Sum of marks in 4 subjects + Marks in the 5th subject 5 = = = 78 Theref ore, her average in all 5 subjects is 78.