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Let F be the class of all intervals of the form (a, b) where a lessthan b and a, b epsilon R are...

Question:

Let F be the class of all intervals of the form (a, b) where a < b and a, b {eq}\epsilon {/eq} R are arbitrary. Show that {eq}\sigma(F) = B {/eq}.

If P(A) = 1/3 and P(B{eq}^c {/eq}) = 1/4, are A and B be mutually exclusive? Prove your answer.

Mutually Exclusive Events:

The mutual exclusive events can not occur simultaneously. The joint events are known as mutually exclusive events. The intersection between the event A and the event B will be zero.

Answer and Explanation:

Given that,

{eq}P(A) = \dfrac{1}{3}\\ P(B^c) = \dfrac{1}{4} {/eq}


Now,

{eq}P(B) = 1 - P(B^c)\\ P(B) = 1 - \dfrac{1}{4}\\ P(B) = \dfrac{3}{4} {/eq}


If event A and event B are mutually exclusive then,

{eq}P(A \ o \ B)= P(A) + P(B) = 1 {/eq}


Now,

{eq}P(A) + P(B) = \dfrac{1}{3} + \dfrac{3}{4}\\ P(A) + P(B) = \dfrac{7}{12} \ne 1 {/eq}


Therefore, events A and B are not mutually exclusive.


Learn more about this topic:

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Mutually Exclusive Events & Non-Mutually Exclusive Events

from 6th-8th Grade Math: Practice & Review

Chapter 48 / Lesson 5
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