A die is rolled. Let E be the event “die shows 4” and F be the event “die shows even number”. Are E and F mutually exclusive?
No.
A die is thrown. Describe the following events:
(i) A: a number less than 7
(ii) B: a number greater than 7
(iii) C: a multiple of 3
(iv) D: a number less than 4
(v) E: an even number greater than 4
(vi) F: a number not less than 3
Also find A ∪ B, A ∩ B, B ∪ C, E ∩ F, D ∩ E, A − C, D − E, E ∩ F′, F′.
(i) A = {1, 2, 3, 4, 5, 6}
(ii) B = ∅
(iii) C = {3, 6}
(iv) D = {1, 2, 3}
(v) E = {6}
(vi) F = {3, 4, 5, 6}
A ∪ B = {1, 2, 3, 4, 5, 6}
A ∩ B = ∅
B ∪ C = {3, 6}
E ∩ F = {6}
D ∩ E = ∅
A − C = {1, 2, 4, 5}
D − E = {1, 2, 3}
E ∩ F′ = ∅
F′ = {1, 2}
An experiment involves rolling a pair of dice. Describe the events:
A: the sum is greater than 8
B: 2 occurs on either die
C: the sum is at least 7 and a multiple of 3
Which pairs of these events are mutually exclusive?
A = {(3,6), (4,5), (5,4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)}
B = {(1,2), (2,2), (3,2), (4,2), (5,2), (6,2), (2,1), (2,3), (2,4), (2,5), (2,6)}
C = {(3,6), (6,3), (5,4), (4,5), (6,6)}
A and B, B and C are mutually exclusive.
Three coins are tossed once. Let A denote the event ‘three heads show’, B denote the event ‘two heads and one tail show’, C denote the event ‘three tails show’ and D denote the event ‘a head shows on the first coin’. Determine whether the events are:
(i) mutually exclusive
(ii) simple
(iii) compound
(i) A and B; A and C; B and C; C and D are mutually exclusive.
(ii) A and C are simple.
(iii) B and D are compound.
Three coins are tossed. Describe:
(i) Two events which are mutually exclusive.
(ii) Three events which are mutually exclusive and exhaustive.
(iii) Two events which are not mutually exclusive.
(iv) Two events which are mutually exclusive but not exhaustive.
(v) Three events which are mutually exclusive but not exhaustive.
(i) “Getting at least two heads” and “getting at least two tails”
(ii) “Getting no heads”, “getting exactly one head” and “getting at least two heads”
(iii) “Getting at most two tails” and “getting exactly two tails”
(iv) “Getting exactly one head” and “getting exactly two heads”
(v) “Getting exactly one tail”, “getting exactly two tails”, “getting exactly three tails”
Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤ 5.
Describe the events:
(i) A′
(ii) not B
(iii) A or B
(iv) A and B
(v) A but not C
(vi) B or C
(vii) B and C
(viii) A ∩ B′ ∩ C′
(i) A′ = B
(ii) B′ = A
(iii) A ∪ B = S
(iv) A ∩ B = ∅
(v) A − C = {(2,4), (2,5), (2,6), (4,2), (4,3), (4,4), (4,5), (4,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}
(vi) B ∪ C = {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)}
(vii) B ∩ C = {(1,1),(1,2),(1,3),(1,4),(3,1),(3,2)}
(viii) A ∩ B′ ∩ C′ = {(2,4),(2,5),(2,6),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}
Referring to question 6, state true or false (with reasons):
(i) A and B are mutually exclusive
(ii) A and B are mutually exclusive and exhaustive
(iii) A = B′
(iv) A and C are mutually exclusive
(v) A and B′ are mutually exclusive
(vi) A′, B′, C are mutually exclusive
(i) True
(ii) True
(iii) True
(iv) False
(v) False
(vi) False
Which of the following cannot be valid assignments of probabilities for outcomes of the sample space \(S = \{\omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}, \omega_{5}, \omega_{6}, \omega_{7}\}\)?
Assignments:
(a) \(P(\omega_{1}) = 0.1, P(\omega_{2}) = 0.01, P(\omega_{3}) = 0.05, P(\omega_{4}) = 0.03, P(\omega_{5}) = 0.01, P(\omega_{6}) = 0.2, P(\omega_{7}) = 0.6\)
(b) \(P(\omega_{1}) = P(\omega_{2}) = P(\omega_{3}) = P(\omega_{4}) = P(\omega_{5}) = P(\omega_{6}) = P(\omega_{7}) = 1/7\)
(c) \(P(\omega_{1}) = 0.1, P(\omega_{2}) = 0.2, P(\omega_{3}) = 0.3, P(\omega_{4}) = 0.4, P(\omega_{5}) = 0.5, P(\omega_{6}) = 0.6, P(\omega_{7}) = 0.7\)
(d) \(P(\omega_{1}) = -0.1, P(\omega_{2}) = 0.2, P(\omega_{3}) = 0.3, P(\omega_{4}) = 0.4, P(\omega_{5}) = -0.2, P(\omega_{6}) = 0.1, P(\omega_{7}) = 0.3\)
(e) \(P(\omega_{1}) = 1/14, P(\omega_{2}) = 2/14, P(\omega_{3}) = 3/14, P(\omega_{4}) = 4/14, P(\omega_{5}) = 5/14, P(\omega_{6}) = 6/14, P(\omega_{7}) = 15/14\)
(a) Yes
(b) Yes
(c) No
(d) No
(e) No
A coin is tossed twice. What is the probability that at least one tail occurs?
3/4
A die is thrown. Find the probability of the following events:
(i) A prime number will appear.
(ii) A number greater than or equal to 3 will appear.
(iii) A number less than or equal to 1 will appear.
(iv) A number more than 6 will appear.
(v) A number less than 6 will appear.
(i) 1/2
(ii) 2/3
(iii) 1/6
(iv) 0
(v) 5/6
A card is selected from a pack of 52 cards.
(a) How many points are there in the sample space?
(b) Calculate the probability that the card is an ace of spades.
(c) Calculate the probability that the card is (i) an ace (ii) a black card.
(a) 52
(b) 1/52
(c) (i) 1/13 (ii) 1/2
A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed. Find the probability that the sum of the numbers that turn up is
(i) 3
(ii) 12.
(i) 1/12
(ii) 1/12
There are four men and six women on the city council. If one council member is selected for a committee at random, how likely is it that it is a woman?
3/5
A fair coin is tossed four times, and a person wins Rs 1 for each head and loses Rs 1.50 for each tail that turns up.
From the sample space, calculate how many different amounts of money you can have after four tosses and the probability of having each of these amounts.
Possible amounts: Rs 4.00 gain, Rs 1.50 gain, Re 1.00 loss, Rs 3.50 loss, Rs 6.00 loss.
P(Winning Rs 4.00) = 1/16
P(Winning Rs 1.50) = 1/4
P(Losing Re 1.00) = 3/8
P(Losing Rs 3.50) = 1/4
P(Losing Rs 6.00) = 1/16
Three coins are tossed once. Find the probability of getting
(i) 3 heads
(ii) 2 heads
(iii) at least 2 heads
(iv) at most 2 heads
(v) no head
(vi) 3 tails
(vii) exactly two tails
(viii) no tail
(ix) at most two tails.
(i) 1/8
(ii) 3/8
(iii) 1/2
(iv) 7/8
(v) 1/8
(vi) 1/8
(vii) 3/8
(viii) 1/8
(ix) 7/8
If 2/11 is the probability of an event, what is the probability of the event ‘not A’?
9/11
A letter is chosen at random from the word “ASSASSINATION”. Find the probability that the letter chosen is
(i) a vowel
(ii) a consonant.
(i) 6/13
(ii) 7/13
In a lottery, a person chooses six different natural numbers at random from 1 to 20 and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game? (Order of the numbers is not important.)
1/38760
Check whether the following probabilities P(A) and P(B) are consistently defined:
(i) P(A) = 0.5, P(B) = 0.7, P(A ∩ B) = 0.6
(ii) P(A) = 0.5, P(B) = 0.4, P(A ∪ B) = 0.8
(i) No, because P(A ∩ B) must be less than or equal to P(A) and P(B).
(ii) Yes
Fill in the blanks in the following table:
P(A), P(B), P(A ∩ B), P(A ∪ B):
(i) P(A) = 1/3, P(B) = 1/5, P(A ∩ B) = 1/15, P(A ∪ B) = ?
(ii) P(A) = 0.35, P(B) = ?, P(A ∩ B) = 0.25, P(A ∪ B) = 0.6
(iii) P(A) = 0.5, P(B) = 0.35, P(A ∩ B) = ?, P(A ∪ B) = 0.7
(i) 7/15
(ii) 0.5
(iii) 0.15
Given P(A) = 3/5 and P(B) = 1/5. Find P(A or B), if A and B are mutually exclusive events.
4/5
If E and F are events such that P(E) = 1/4, P(F) = 1/2 and P(E and F) = 1/8, find
(i) P(E or F)
(ii) P(not E and not F).
(i) 5/8
(ii) 3/8
Events E and F are such that P(not E or not F) = 0.25. State whether E and F are mutually exclusive.
No
A and B are events such that P(A) = 0.42, P(B) = 0.48 and P(A and B) = 0.16. Determine
(i) P(not A)
(ii) P(not B)
(iii) P(A or B).
(i) 0.58
(ii) 0.52
(iii) 0.74
In Class XI of a school, 40% of the students study Mathematics and 30% study Biology. 10% of the class study both Mathematics and Biology. If a student is selected at random from the class, find the probability that the student will be studying Mathematics or Biology.
0.6
In an entrance test graded on the basis of two examinations, the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7. The probability of passing at least one of them is 0.95. What is the probability of passing both?
0.55
The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither is 0.1. If the probability of passing the English examination is 0.75, what is the probability of passing the Hindi examination?
0.65
In a class of 60 students, 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random, find the probability that
(i) the student opted for NCC or NSS
(ii) the student has opted neither NCC nor NSS
(iii) the student has opted NSS but not NCC.
(i) 19/30
(ii) 11/30
(iii) 2/15
A box contains 10 red marbles, 20 blue marbles and 30 green marbles. Five marbles are drawn from the box, what is the probability that
(i) all will be blue?
(ii) at least one will be green?
(i) \(\dfrac{^{20}C_{5}}{^{60}C_{5}}\)
(ii) \(1-\dfrac{^{30}C_{5}}{^{60}C_{5}}\)
Four cards are drawn from a well–shuffled deck of 52 cards. What is the probability of obtaining 3 diamonds and 1 spade?
\(\dfrac{^{13}C_{3}\cdot{}^{13}C_{1}}{^{52}C_{4}}\)
A die has two faces each with number 1, three faces each with number 2 and one face with number 3. If the die is rolled once, determine
(i) \(P(2)\)
(ii) \(P(1\text{ or }3)\)
(iii) \(P(\text{not }3)\)
(i) \(\dfrac{1}{2}\)
(ii) \(\dfrac{1}{2}\)
(iii) \(\dfrac{5}{6}\)
In a certain lottery 10,000 tickets are sold and ten equal prizes are awarded. What is the probability of not getting a prize if you buy
(a) one ticket
(b) two tickets
(c) ten tickets?
(a) \(\dfrac{999}{1000}\)
(b) \(\dfrac{^{9990}C_{2}}{^{10000}C_{2}}\)
(c) \(\dfrac{^{9990}C_{10}}{^{10000}C_{10}}\)
Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among the 100 students, what is the probability that
(a) you both enter the same section?
(b) you both enter different sections?
(a) \(\dfrac{17}{33}\)
(b) \(\dfrac{16}{33}\)
Three letters are dictated to three persons and an envelope is addressed to each of them. The letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope.
\(\dfrac{2}{3}\)
A and B are two events such that \(P(A)=0.54\), \(P(B)=0.69\) and \(P(A\cap B)=0.35\). Find
(i) \(P(A\cup B)\)
(ii) \(P(A'\cap B')\)
(iii) \(P(A\cap B')\)
(iv) \(P(B\cap A')\).
(i) 0.88
(ii) 0.12
(iii) 0.19
(iv) 0.34
From the employees of a company, 5 persons are selected to represent them in the managing committee of the company. The particulars of these five persons are:
Harish (M, 30 years), Rohan (M, 33 years), Sheetal (F, 46 years), Alis (F, 28 years), Salim (M, 41 years).
A person is selected at random from this group to act as a spokesperson. What is the probability that the spokesperson will be either male or over 35 years?
\(\dfrac{4}{5}\)
If 4-digit numbers greater than 5000 are randomly formed from the digits 0, 1, 3, 5 and 7, what is the probability of forming a number divisible by 5 when
(i) the digits are repeated?
(ii) the repetition of digits is not allowed?
(i) \(\dfrac{33}{83}\)
(ii) \(\dfrac{3}{8}\)
The number lock of a suitcase has 4 wheels, each labelled with ten digits, i.e. from 0 to 9. The lock opens with a sequence of four digits with no repeats. What is the probability of a person getting the right sequence to open the suitcase?
\(\dfrac{1}{5040}\)