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

Question

Mathematics

Carleen

Yesterday, 22:57

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

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Annabel Jacobson · Answer 1

Not mutually exclusive

Step-by-step explanation:

In the rule of probability, for two events to be mutually exclusive, the probability of them occuring at the same time must be 0.

In this case P (true and false) = 0 to be mutually exclusive.

In the term of Vann diagram 'true and false' in this case represent the intersection of two events 'true' and 'false'. And if it is shown in Vann diagram, both group will be apart from each other or the value inside the Vann diagram is 0.

Therefore,

We know from statistics that

P (A and B) = P (A) + P (B) - P (A or B)

Translating into this case

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

= 0.9 + 0.82 - 0.75 = 0.97

Therefore, this event is not mutually exclusive.