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## Sunday 21 February 2021

### CBSE Class 11 Maths - MCQ and Online Tests - Unit 16 - Probability

#### CBSE Class 11 Maths  – MCQ and Online Tests – Unit 16 – Probability

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#### CBSE Class 11 Maths  – MCQ and Online Tests – Unit 16 – Probability

Question 1.
A certain company sells tractors which fail at a rate of 1 out of 1000. If 500 tractors are purchased from this company, what is the probability of 2 of them failing within first year
(a) e-1/2/2
(b) e--1/2/4
(c) e-1/2/8
(d) none of these

Hint:
This question is based on Poisson distribution.
Now, ? = np = 500в(1/1000) = 500/1000 = 1/2
Now, P(x = 2) = {e-1/2 в (1/2)╡}/2! = e-1/2 /(4в2) = e-1/2/8

Question 2.
A random variable X has poison distribution with mean 2. Then, P (X > 1.5) equals
(a) 1 – 3/e╡
(b) 2/e╡
(c) 3/e╡
(d) 0

Hint:
Here m = 2
Now, P(X > 1.5) = ?r {(e-2 в 2r)/r!} {2 = r = 8}
= e-2 {2╡/2! + 2Ё/3! + 24/4! + …}
= e-2 {(1 + 2 /1! + 2╡/2! + 2Ё/3! + …) – 1 – 2}
= e-2 (e╡ – 3)
= 1 – 3e-2
= 1 – 3/e╡

Question 3.
Three identical dice are rolled. The probability that the same number will appear on each of them is
(a) 1/6
(b) 1/36
(c) 1/18
(d) 3/28

Hint:
Total number of cases = 6Ё = 216
The same number can appear on each of the dice in the following ways:
(1, 1, 1), (2, 2, 2), ………….(3, 3, 3)
So, favourable number of cases = 6
Hence, required probability = 6/216 = 1/36

Question 4.
There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, in a random order till both the faulty machines are identified. Then the probability that only two tests are needed is
(a) 1/3
(b) 1/6
(c) 1/2
(d) 1/4

Hint:
First, we choose 1 machine out of given 4.
The probability that it is fault = 2/4 = 1/2
Now, we have to pick the second fault machine.
The probability that it is fault = 1/3
So, required probability = (1/2)в(1/3) = 1/6

Question 5.
Two unbiased dice are thrown. The probability that neither a doublet nor a total of 10 will appear is
(a) 3/5
(b) 2/7
(c) 5/7
(d) 7/9

Hint:
When two dice are throw, then Total outcome = 36
A doublet: {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
Favourable outcome = 6
Sum is 10: {(4, 6), (5, 5), (6, 4)}
Favourable outcome = 3
Again, A doublet and sum is 10: (5, 5)
Favourable outcome = 1
Now, P(either dublet or a sum of 10 appears) = P(A dublet appear) + P(sum is 10) – P(A dublet appear and sum is 10)
? P(either dublet or a sum of 10 appears) = 6/36 + 3/36 – 1/36
= (6 + 3 – 1)/36
= 8/36
= 2/9
So, P(neither dublet nor a sum of 10 appears) = 1 – 2/9 = 7/9

Question 6.
Two dice are thrown the events A, B, C are as follows A: Getting an odd number on the first die. B: Getting a total of 7 on the two dice. C: Getting a total of greater than or equal to 8 on the two dice. Then AUB is equal to
(a) 15
(b) 17
(c) 19
(d) 21

Hint:
When two dice are thrown, then total outcome = 6в6 = 36
A: Getting an odd number on the first die.
A = {(1, 1), (1, 2), (1, 3), (1, 4),(1, 5), (1, 6), (3, 1), (3, 2), (3, 3), (3, 4),(3, 5), (3, 6), (5, 1), (5, 2), (5, 3), (5, 4),(5, 5), (5, 6)}
Total outcome = 18
B: Getting a total of 7 on the two dice.
B = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
Total outcome = 6
C: Getting a total of greater than or equal to 8 on the two dice.
C = {(2, 6), (3, 5), (3, 6), (4, 4),(4, 5), (4, 6), (5, 3), (5, 4), (5, 5), (5, 6),(6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
Total outcome = 15
Now n(A ? B) = n(A) + n(B) – n(A n B)
? n(A ? B) = 18 + 6 – 3
? n(A ? B) = 21

Question 7.
Two numbers are chosen from {1, 2, 3, 4, 5, 6} one after another without replacement. Find the probability that the smaller of the two is less than 4.
(a) 4/5
(b) 1/15
(c) 1/5
(d) 14/15

Hint:
Total number of ways of choosing two numbers out of six = 6C2 = (6в5)/2 = 3в5 = 15
If smaller number is chosen as 3 then greater has choice are 4, 5, 6
So, total choices = 3
If smaller number is chosen as 2 then greater has choice are 3, 4, 5, 6
So, total choices = 4
If smaller number is chosen as 1 then greater has choice are 2, 3, 4, 5, 6
So, total choices = 5
Total favourable case = 3 + 4 + 5 = 12
Now, required probability = 12/15 = 4/5

Question 8.
The probability that when a hand of 7 cards is drawn from a well-shuffled deck of 52 cards, it contains 3 Kings is
(a) 1/221
(b) 5/716
(c) 9/1547
(d) None of these

Hint:
Total number of cards = 52
Number of king card = 4
Now, 7 cards are drawn from 52 cards.
P (3 cards are king) = {4C3 в 48C4}/52C7
= {4в(48в47в46в45)/(4в3в2в1)}/{(52в51в50в49в48в47в46)/(7в6в5в4в3в2в1)}
= {4в(48в47в46в45)в(7в6в5в4в3в2в1)}/{(4в3в2в1)в{(52в51в50в49в48в47в46)}
= (7в6в5в4в45)/(52в51в50в49)
= (6в5в4в45)/(52в51в50в7)
= (6в4в45)/(7в52в51в10)
= (6в45)/(7в13в51в10)
= (6в3)/(7в13в17в2)
= (3в3)/(7в13в17)
= 9/1547

Question 9.
A certain company sells tractors which fail at a rate of 1 out of 1000. If 500 tractors are purchased from this company, what is the probability of 2 of them failing within first year
(a) e-1/2/2
(b) e-1/2/4
(c) e-1/2/8
(d) none of these

Hint:
This question is based on Poisson distribution.
Now, ? = np = 500в(1/1000) = 500/1000 = 1/2
Now, P(x = 2) = {e-1/2 в (1/2)╡}/2! = e-1/2/(4в2) = e-1/2/8

Question 10.
The probability that in a random arrangement of the letters of the word INSTITUTION the three T are together is
(a) 0.554
(b) 0.0554
(c) 0.545
(d) 0.0545

Hint:
Given word: INSTITUTION
Total letters = 11
The word contains 3 I, 2 N, 1 S, 3 T, 1 U and 1 O
Total number of arrangement = 11!/(3!в2!в3!) = 554400
Now, taken 3 T are together.
So total latter = 9
The number of favorable cases = 9!/(3!в2!) = 30240
Now, P(3 T are together) = 30240/554400 = 0.0545

Question 11.
Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is
(a) 2/9
(b) 1/9
(c) 8/9
(d) 7/9

Hint:
One person can select one house out of 3 = 3C1 = 3
So, three persons can select one house out of three = 3в3в3 = 27
Thus, probability that all the three can apply for the same house = 3/27 = 1/9

Question 12.
Two cards from a pack of 52 cards are lost. One card is drawn from the remaining cards. If drawn card is diamond then the probability that the lost cards were both hearts is
(a) 143/1176
(b) 143/11760
(c) 143/11706
(d) 134/11760

Hint:
Total number of cards = 52
Two cards are lost.
So remaining cards = 50
Now one card is drawn.
Probability that it is a diamond card = 13/50
Now probability that both lost cards are heart = 13/50 в(11C2 / 49C2)
= 13/50 в[{(11в10)/2}/{(49в48/2)}]
= 13/50 в{(11в10)/(49в48)}
= {(13в11в10)/(50в49в48)}
= {(13в11)/(5в49в48)}
= 143/11760
So probability that both lost card are heart = 143/11760

Question 13.
If four whole numbers taken at random are multiplied together, then the chance that the last digit in the product is 1, 3, 5, 7 is
(a) 16/25
(b) 16/125
(c) 16/625
(d) none of these

Hint:
The last digit of the four whole number can be
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
The chance that any of the four numbers is divisible by 2 or 5 = 6/10 = 3/5
Hence, the chance that any of the four numbers is not divisible by 2 or 5 = 1 – 3/5 = 2/5
So, the chance that all of the four numbers are divisible by 2 or 5 = (2/5)в(2/5)в(2/5)в(2/5)
= 16/625
This is the chance that the last digit in the product will not be 0, 2, 4, 5, 6, 8 and this is also the chance that the last digit in the product is 1, 3, 7 or 9

Question 14.
A couple has two children. The probability that both children are females if it is known that the elder child is a female is
(a) 0
(b) 1
(c) 1/2
(d) 1/3

Hint:
Given, a couple has two children.
Let A denotes both children are females i.e. {FF}
Now, P(A) = (1/2)в(1/2) = 1/4
and B denotes elder children is a female i.e. {FF, FM}
P(B) = 1/4 + 1/4 = 1/2
Now, P(A n B) = 1/4
Now, P(Both the children are female if elder child is female)
P(A/B) = P(A n B)/P(B)
? P(A/B) = (1/4)/(1/2)
? P(A/B) = 1/2

Question 15.
The probability of getting the number 6 at least once in a regular die if it can roll it 6 times?
(a) 1 – (5/6)6
(b) 1 – (1/6)6
(c) (5/6)6
(d) (1/6)6

Hint:
Let A is the event that 6 does not occur at all.
Now, the probability of at least one 6 occur = 1 – P(A)
= 1 – (5/6)6

Question 16.
On his vacation, Rahul visits four cities (A, B, C, and D) in a random order. The probability that he visits A first and B last is
(a) 1/2
(b) 1/6
(c) 1/10
(d) 1/12

Hint:
Total cities are 4 i.e. A, B, C, D
Given, Rahul visit four cities, So, n(S) = 4! = 24
Now, sample space IS:
Let G = Rahul visits A firsta and B last
? n(G) = 2
So, P(G) = n(G)/n(S) = 2/24 = 1/12

Question 17.
Let A and B are two mutually exclusive events and if P(A) = 0.5 and P(B ╞) = 0.6 then P(A?B) is
(a) 0
(b) 1
(c) 0.6
(d) 0.9

Hint:
Given, A and B are two mutually exclusive events.
So, P(A n B) = 0
Again given P(A) = 0.5 and P(B ╞) = 0.6
P(B) = 1 – P(B ╞) = 1 – 0.6 = 0.4
Now, P(A ? B) = P(A) + P(B) – P(A n B)
? P(A ? B) = P(A) + P(B)
? P(A ? B) = 0.5 + 0.4 = 0.9

Question 18.
The probability of getting 53 Sundays in a leap year is
(a) 1/7
(b) 2/7
(c) 3/7
(d) None of these

Hint:
In a leap year, the total number of days = 366 days.
In 366 days, there are 52 weeks and 2 days.
Now two days may be
(i) Sunday and Monday
(ii) Monday and Tuesday
(iii) Tuesday and Wednesday
(iv) Wednesday and Thursday
(v) Thursday and Friday
(vi) Friday and Saturday
(vii) Saturday and Sunday
Now there are total 7 possibilities, So total outcomes = 7
In 7 possibilities, Sunday came two times.
So, favorable case = 2
Hence, the probabilities of getting 53 Sundays in a leap year = 2/7

Question 19.
A bag contains 5 brown and 4 white socks . A man pulls out two socks. The probability that both the socks are of the same colour is
(a) 9/20
(b) 2/9
(c) 3/20
(d) 4/9

Hint:
Total number of shocks = 5 + 4 = 9
Two shocks are pulled.
Now, P(Both are same color) = (5C2 + 4C2)/9C2
= {(5в4)/(2в1) + (4в3)/(2в1)}/{(9в8)/(2в1)}
= {(5в4) + (4в3)/}/{(9в8)
= (5 + 3)/(9в2)
= 8/18
= 4/9

Question 20.
When a coin is tossed 8 times getting a head is a success. Then the probability that at least 2 heads will occur is
(a) 247/265
(b) 73/256
(c) 247/256
(d) 27/256

Hint:
Let x be number a discrete random variable which denotes the number of heads obtained in n (in this question n = 8)
The general form for probability of random variable x is
P(X = x) = nCx в px в qn-x
Now, in the question, we want at least two heads
Now, p = q = 1/2
So, P(X = 2) = 8C2 в (1/2)╡ в (1/2)8-2
? P(X = 2) = 8C2 в (1/2)╡ в (1/2)6
? 1 – P(X < 2) = 8C0 в (1/2)0 в (1/2)8 + 8C1 в (1/2)1 в (1/2)8-1
? 1 – P(X < 2) = (1/2)8 + 8 в (1/2)1 в (1/2)7
? 1 – P(X < 2) = 1/256 + 8 в (1/2)8
? 1 – P(X < 2) = 1/256 + 8/256
? 1 – P(X < 2) = 9/256
? P(X < 2) = 1 – 9/256
? P(X < 2) = (256 – 9)/256
? P(X < 2) = 247/256

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