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प्रश्न
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
- A'
- not B
- A or B
- A and B
- A but not C
- B or C
- B and C
- A ∩ B' ∩ C'
उत्तर
Sample space when two coins are tossed
S = {(1, 1), (1, 2), …(1, 6), (2, 1), (2, 2), …(2, 6), (3, 1), (3, 2), …(3, 6), (4, 1), (4, 2), …(4, 6), (5, 1), (5, 2), …(5, 6), (6, 1), …(6, 6)}
A = Getting an even number on the first die.
= {(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 = Getting an odd number on the first die.
= {(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
= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}
i. A’ = S – 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)}
= B
ii. Not B = B’ = No odd number on the first die
= {(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
iii. A or B = A ∪ B = {x : x is an even number on the first die} ∪ {is an odd number on the first die}
= S
iv. A and B = A ∩ B
= {x : x is an even number on the first die} ∩ {x is an odd number on the first die}
= ϕ
v. A but not C
= {x : x is an even number on the first dice} – {sum of numbers obtained on the dice ≤ 5}
A – C = {(2, 1), (2, 2), …(2, 6), (4, 1), (4, 2), …(4, 2), …(4, 6), (6, 1), (6, 2), ….(6, 6)} – {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}
= {(2, 4), (2, 5), (2, 6), (4, 2), (4, 3), …(4, 6), (6, 1), (6, 2), .…(6, 6)}
vi. B or C = B ∪ C = {x : x will be an odd number on the first die} ∪ {Sum of numbers obtained on dice ≤ 5}
= {(1, 1), (1, 2), …(1, 6), (3, 1), (3, 2), …(3, 6), (5, 1), (5, 2), …(5, 6)} ∪ (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 2), (4, 1)}
= {(1, 1), (1, 2), …(1, 6), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), …(3, 6), (4, 1), (5, 1), (5, 2), (5, 3), …(5, 6).
viii. B and C i.e. B ∩ C = {(1, 1), …(1, 6), (3, 1), (3, 2), …(3, 6), (5, 1), (5, 2), (5, 3), …(5, 6) ∩ {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2) (2, 3), (3, 1), (3, 2), (4, 1)}
= {(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)}
viii. Here, B’ = A
∴ A ∩ B’ = A ∩ A = A
∴ A ∩ B’ ∩ C’ = {(2, 1), (2, 2), …(2, 6), (4, 1), (4, 2), …(4, 6), (6, 1), (6, 2), …(6, 6)} ∩ {(1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6), (4, 2), (4, 3), …(4, 6), (5, 1), (5, 2), …(5, 6), (6, 1), (6, 2), …(6, 5)}
= {(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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