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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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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?
Q. Three coins are tossed
i. Describe two events which are mutually exclusive.
ii. Describe three events which are mutually exclusive and exhaustive.
iii. Describe two events, which are not mutually exclusive.
iv. Describe two events which are mutually exclusive but not exhaustive.
v. Describe three events which are mutually exclusive but not exhaustive.

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