Question

LetA be the probability of not wearing glasses and B be the probability of having brown eyes. Suppose both events are independent such that P ( A ) = 0.90  and P ( B ) = 0.70. We want to find what proportion are not wearing glasses or have brown eyes (or both). This means we must find P ( A ∪ B ).

In this case, we would use our union formula:

P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) = 0.90 + 0.70 − 0.63 = 0.97

Hence, P ( A ∪ B ) = 0.97 (try it yourself before proceeding). Please create your own probability of the union example and explain why subtraction is needed in the union formula. In other words, why would it be incorrect to just simply add the probabilities outright? Relate the example to your major. (Mayor is nursing)

Answer 1

Suppose we wish to determine the probability of majoring in Nursing and the student being a female.

Suppose Pr of a nursing student being female is 0.50

Let N = Probabilty that student is majoring in Nursing F = Probabilty that student is female

We have to find the probability P(NUF)

Suppose, we simply add the probabilities outright: From the above diagram,

Pr(N) = Pr(NURSING MAJOR) + Pr(FEMALE NURSING STUDENT)

= 0.80

Pr(F) = Pr(FEMALE) + Pr(FEMALE NURSING STUDENT)

= 0.60

Now,

Pr(N) + Pr(F)

= [Pr(NURSING MAJOR) + Pr(FEMALE NURSING STUDENT)] + [Pr(FEMALE) + Pr(FEMALE NURSING STUDENT)]

= 0.80 + 0.60

= 1.40

We find that the probability exceeds unity.We also notice that the probability of the event "Pr(FEMALE NURSING STUDENT)]" is counted twice.

Hence, we need to subtact one of the repeated probabilities Pr(FEMALE NURSING STUDENT)] to obtain the correct probability P(NUF):

Hence,

= 0.80 + 0.60 - 0.50

= 0.90