In this sample, are the events "believed the false headline" and "believed the true headline" mutually exclusive?

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% and the probability that the student believed the false headline was 82%. She found that 75% of the students believed both headlines.

Find the probability that a randomly selected person from this sample believed the true headline OR believed the false headline.

Question:

In this sample, are the events "believed the false headline" and "believed the true headline" mutually exclusive?

Answer:

No

Answer: The events "believed the false headline" and "believed the true headline" are not mutually exclusive.

The probability that a randomly selected person from this sample believed the true headline OR believed the false headline is 0.97.

Step-by-step explanation:

Let T = Event "believed the true headline"

F = Event "believed the false headline"

As per given we have

P (T) = 0.90

P (F) = 0.82

P (T and F) = 0.75

The events "believed the false headline" and "believed the true headline" are not mutually exclusive ∵ P (T and F) ≠ 0.

Also, the probability that a randomly selected person from this sample believed the true headline OR believed the false headline would be

P (T or F) = P (T) + P (F) - P (T and F)

= 0.90 + 0.82 - 0.75

=0.97

Therefore, the probability that a randomly selected person from this sample believed the true headline OR believed the false headline is 0.97.