A civics teacher asked her students to indicate whether they believed each of two headlines. One headline was false and the other was true, but the students did not know this. The probability that a student selected at random believed the true headline was 90\%90%90, percent and the probability that the student believed the false headline was 82\%82%82, percent. She found that 75\%75%75, percent of the students believed both headlines. In this sample, are the events "believed the false headline" and "believed the true headline" mutually exclusive?.

For two occasions, let say E₁ and E₂ are supposed to be fundamentally unrelated in the event that the two of them can't happen all the while. In set-hypothetical documentation, we can say that the two sets E₁ and E₂ are disjoint. for example E₁ ∩ E₂ = ∅ and the likelihood of them happening simultaneously is zero. for example P( E₁ ∩ E₂ ) = 0

In the probability, for two occasions to be fundamentally unrelated, the likelihood of them happening simultaneously should be 0.

For this situation P(true and bogus) = 0 to be fundamentally unrelated.

In term of Venn diagram 'valid and misleading' for this situation address the convergence of two occasions 'valid' and 'bogus'. What's more, assuming it is displayed in Vann outline, both gathering will be separated from one another or the worth inside the Vann graph is 0.

Consequently,

We know from measurements that

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Converting into this case

P(true or false) = P(true) + P(false) - P(true and false)

= 0.9 + 0.82 - 0.75 = 0.97

Here, we can see that P(true or false) is greater than 0,it means that the events are not mutually exclusive.

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