Question

what are the probabilities of the events A and B if a single dice is thrown twice in a row:

A= “ the sum of all the results is an even number”

B= “ the result of the second throw is an even number”

need to calculate probabilities of both events to then answer if the events are independent using the rule of multiplication of independent events.

Answer 1

When a dice is thrown twice, the following possible samples may arise:
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)}

The samples favourable to A are:
A = {(1,1), (1,3), (1,5),
(2,2), (2,4), (2,6),
(3,1),(3,3), (3,5),
​​​(4,2), (4,4), (4,6),
(5,1), (5,3), (5,5),
(6,2), (6,4), (6,6)}
​​​​​​​
​​​​​​​Now,

The samples favourable to B are:
B = {(1,2), (1,4), (1,6),
(2,2), (2,4) (2,6),
(3,2), (3,4), (3,6),
​​​(4,2), (4,4), (4,6),
(5,2), (5,4), (5,6),
​(6,2), (6,4), (6,6)}

​​​​​​​Now,
​​​​​​​

The samples favourable to both A and B are:

= {(2,2), (2,4) (2,6),
​​​(4,2), (4,4), (4,6),
​(6,2), (6,4), (6,6)}

​​​​​​​Now,

For the events A and B to be independent, we must have,

R.H.S.




= L.H.S.

Hence, the events are independent using the rule of multiplication of independent events.

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