One of my kids frequently tries to get his way by asking for permission for a tr

Question

One of my kids frequently tries to get his way by asking for permission for a treat from “Mom” or “Dad.” Let M be the event that the Mom said yes and D be the event that the Dad says yes. Suppose that M and D are independent events with P(A)=.4 and P(B) = .7

a) Given Mom says “No”, what is the probability that Dad also says No? Explain your reasoning AND Write out the probability in mathematical symbols as well as calculate probability.
b)What is the probability that both Mom and Dad will say yes? Write out the probability in mathematical symbols as well as calculate probability

Explanation / Answer

a) Given that mom says NO, probability that Dada also says no is computed as:

P( Bc | Ac ) = P( Ac and Bc ) / P(Ac )

Now from the definition of the complement of the set, we get :P(Ac ) = 1- P(A) = 1 - 0.4 = 0.6

P(Ac and Bc) = 1 - P(A or B)

Using law of total probability, we get:

P(A or B) = P(A) + P(B) - P(A and B) = P(A) + P(B) - P(A)P(B) = 0.4 + 0.7 - 0.4*0.7 = 0.82

Therefore, we get: P(Ac and Bc) = 1 - P(A or B) = 1 - 0.82 = 0.18

Therefore, we get:

P( Bc | Ac ) = P( Ac and Bc ) / P(Ac ) = 0.18 / 0.6 = 0.3

Therefore 0.3 is the required probability here.

b) Here we have to compute,

probability that both Mom and Dad will say yes. Using the law of independent events, we get:

P(A and B) = P(A)P(B) = 0.4*0.7 = 0.28

Therefore 0.28 is the required probability here.

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