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Standard X
Mathematics
Experimental Probability
Two dice are ...
Question

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

i. Describe the event A′

ii. Describe the event not B

iii. Describe the event A or B

iv. Describe the event A and B

v. Describe the event A but not C

vi. Describe the event B or C

vii. Describe the event B and C

viii. Describe the event A ∩B′ ∩ C′

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Solution

Given: Two dice are thrown
A: getting an even number on the first die.
"If two dice are thrown, then the sample space is"
S=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭

A=⎧⎪⎨⎪⎩(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪⎬⎪⎭

A′=S−A

∴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)⎫⎪⎬⎪⎭


Given: Two dice are thrown
B: getting an odd number on the first die.
"If two dice are thrown, then the sample space is"

S=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭

B=⎧⎪⎨⎪⎩(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)⎫⎪⎬⎪⎭

not B=S−B
=⎧⎪⎨⎪⎩(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪⎬⎪⎭


Given: Two dice are thrown
A: getting an even number on the first die.
B: getting an odd number on the first die.
"If two dice are thrown, then the sample space is"

S=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭

A=⎧⎪⎨⎪⎩(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪⎬⎪⎭

B=⎧⎪⎨⎪⎩(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)⎫⎪⎬⎪⎭

A or B=A ∪ B

=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭

=S


Given: Two dice are thrown
A: getting an even number on the first die.
B: getting an odd number on the first die.
"If two dice are thrown, then the sample space is"

S=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭

A=⎧⎪⎨⎪⎩(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪⎬⎪⎭

B=⎧⎪⎨⎪⎩(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)⎫⎪⎬⎪⎭

∴A and B=A∩B=ϕ


Given: Two dice are thrown
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
"If two dice are thrown, then the sample space is"

S=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭

A=⎧⎪⎨⎪⎩(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪⎬⎪⎭

C: getting the sum of the numbers on the dice ≤5

C={(1,1),(1,2),(1,3),(1,4),(2,1),(2,1),(2,3),(3,1),(3,2),(4,1)}

A but not C
=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),(66)⎫⎪⎬⎪⎭


Given: Two dice are thrown
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
"If two dice are thrown, then the sample space is"

S=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭

B=⎧⎪⎨⎪⎩(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)⎫⎪⎬⎪⎭

C: getting the sum of the numbers on the dice ≤5

C={(1,1),(1,2),(1,3),(1,4),(2,1),(2,1),(2,3),(3,1),(3,2),(4,1)}

B or C=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)⎫⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪⎭

Given: Two dice are thrown
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
"If two dice are thrown, then the sample space is"

S=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭

B=⎧⎪⎨⎪⎩(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)⎫⎪⎬⎪⎭

C: getting the sum of the numbers on the dice ≤5

C={(1,1),(1,2),(1,3),(1,4),(2,1),(2,1),(2,3),(3,1),(3,2),(4,1)}

B and C=B ∩ C

={(1,1),(1,2),(1,3),(1,4),(3,1),(3,2)}


Given: Two dice are thrown
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
"If two dice are thrown, then the sample space is"

S=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭

A=⎧⎪⎨⎪⎩(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪⎬⎪⎭

B: getting an odd number on the first die.

B=⎧⎪⎨⎪⎩(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)⎫⎪⎬⎪⎭

B′=S−B

=⎧⎪⎨⎪⎩(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪⎬⎪⎭

=A

C: getting the sum of the numbers on the dice ≤5

C={(1,1),(1,2),(1,3),(1,4),(2,1),(2,1),(2,3),(3,1),(3,2),(4,1)}

C′=S−C
=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1,5),(1,6),(2,4),(2,5),(2,6),(3,3),(3,4),(3,5),(3,6),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭

B′=A=⎧⎪⎨⎪⎩(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎫⎪⎬⎪⎭

A∩B'∩C'=A∩A∩C'=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)⎫⎪⎬⎪⎭

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Similar questions
Q. 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

i) Describe the event A′
ii) Describe the event not B
iii) Describe the event A or B
iv) Describe the event A and B
v) Describe the event A but not C
vi) Describe the event B or C
vii) Describe the event B and C
viii) Describe the event A∩B′∩C′
Q.

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′

Q. 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′
Q.

Two dice are thrown. The events A, B, C, D, E and F are described as follows :

A = Getting an even number on the first die.

B = Getting an odd number on the first die.

C = Getting at most 5 as sum of the numbers on the two dice.

D = Getting the sum of the numbers on the dice greater than 5 but less than 10.

E = Getting at least 10 as the sum of the numbers on the dice.

F = Getting an odd number on one of the dice.

(i) Describe the following events : A and B, B or C, B and C, A and E, A or F, A and F

(ii) State true or false :

(a) A and B are mutually exclusive.

(b) A and B are mutually exclusive and exhaustive events.

(c) A and C are mutually exclusive events.

(d) C and D are mutually exclusive and exhaustive events.

(e) C, D and E are mutually exclusive and exhaustive events.

(f) A' and B' are mutually exclusive events.

(g) A, B, F arc mutually exclusive and exhaustive events.

Q. Three coins are tossed once. Describe the following events associated with this random experiment:
A = Getting three heads
B = Getting two heads and one tail
C = Getting three tails
D = Getting a head on the first coin.
(i) Which pairs of events are mutually exclusive?
(ii) Which events are elementary events?
(iii) Which events are compound events?
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