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.

The events are not mutually exclusive.

Step-by-step explanation:

To determine that in the sample, are the events "believed the false headline" and "believed the true headline" mutually exclusive.

For two events, let say E₁ and E₂ are said to be mutually exclusive if they cannot both occur simultaneously. In set-theoretical notation, we can say that the two sets  E₁ and E₂ are disjoint. i.e.  E₁ ∩  E₂ = ∅ and the probability of them occurring at the same time is zero. i.e. Pr(  E₁ ∩  E₂ ) = 0

So, we can say;

Let P(true) = E₁  

P(false) = E₂

Thus; statistically:

Pr(  E₁ ∩  E₂ ) = Pr(E₁) + Pr(E₂) - P( E₁  ∪ E₂ )

So;

the probability of students that believed the true headline P(E₁) = 0.90

the probability of students that believed the false headline P(E₁) = 0.82

the probability of students that believed both P( E₁  ∪ E₂ ) = 0.75

∴

Pr(  E₁ ∩  E₂ ) = 0.90 + 0.82 - 0.75

Pr(  E₁ ∩  E₂ ) = 0.97

Thus, the events are not mutually exclusive.

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