In class, it was stated that in order to show that more than two events are independent, you have to check every possible set of intersections. To illustrate this, consider rolling a fair 8-sided die and define the events:A1 = {1, 2, 3, 4} , A2 = {2, 6, 7, 8} and A3 = {1, 2, 3, 5}. Show that the triple intersection behaves as if these events were independent, but none of the pairs of events are independent.

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Asked Oct 3, 2019
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In class, it was stated that in order to show that more than two events are independent, you have to check every possible set of intersections. To illustrate this, consider rolling a fair 8-sided die and define the events:
A1 = {1, 2, 3, 4} , A2 = {2, 6, 7, 8} and A3 = {1, 2, 3, 5}. Show that the triple intersection behaves as if these events were independent, but none of the pairs of events are independent.

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Expert Answer

Step 1

From the given information,

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4 12,3.4 A2,6,7,8 A1,2,3,5

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Step 2

Mutual Independence of three events for any three events A, B and C to be mutually independent the following two conditions must be met:

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P(4n4) P(A) P(4,) P(A4) P(A)xP(4) P(4n4) P(4)x P(4) P(44,04)= P(A)x P (4,) x P (A, )

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Step 3

The triple intersection behaves as if these events were independent, but...

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P(404) 4 P(4)xP(A) 4 8 8 P(44)P(A)xP(4) P(A4) 4 4 P(A,) x P(4) 8 8 P(4n4)P(A)xP(4) 3 P(A4) P(A)xP(4)= 8 8 X X

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