Question

At a certain food establishment the possible food items are rice, chicken, and beef.

Home work help I do not know how to set this up and solve for these. Please show work and box answers. Thank you for your help.

For a given dinner, let R be the event that rice is available, let C be the event that chicken is available, let B be the event that beef is available. Assume that B and C are independent, and that P ( R ) = 0.7, P ( C ) = 0.8, P ( B ) = 0.375, P ( R ∪ C ) = 1, and P ( R | B ) = 0.896.

Find:

P ( R ∪ B ), P ( B ∪ C ), P ( R | C ), P ( C | B ), P ( B n C ), P ( R n C ), P ( R n B), P ( R n C n B), P ( R ∪ C ∪ B ), P ( C | R ), P ( B | C ),  P ( B | R )

Answer 1

We would be looking at the first 8 parts here:

We are given here that:
P(R) = 0.7,
P(C) = 0.8,
P(B) = 0.375,
P(R C) = 1,
P(R | B) = 0.896

a) Using Bayes theorem, we get here:
P( R B) = P(R | B) P(B) = 0.896*0.375 = 0.336
Using law of additivity of probability, we have here:
P( R B) = P(R) + P(B) - P( R B) = 0.7 + 0.375 - 0.336 = 0.739
Therefore 0.739 is the required probability here.

b) P(B C) = P(B) + P(C) - P(B and C)

As B and C are given to be independent, therefore,
P(B and C) = P(B)P(C) = 0.375*0.8 = 0.3

Therefore, P(B C) = 0.375 + 0.8 - 0.3 = 0.875
Therefore 0.875 is the required probability here.

c) First using law of additivity of probability, we have here:
P( R C) = P(R) + P(C) - P( R C) = 0.7 + 0.8 - 1 = 0.5

Using Bayes theorem, we have here:
P(R | C) = P(R C) / P(C) = 0.5/ 0.8 = 0.625
Therefore 0.625 is the required probability here.

d) P(C | B) is again computed using Bayes theorem as:
P(C | B) = P(B and C) / P(B) = 0.3 / 0.375 = 0.8
Therefore 0.8 is the required probability here.

e) P(B C) = already computed in part b) as 0.3.
Therefore 0.3 is the required probability here.

f) First using law of additivity of probability, we have here:
P( R C) = P(R) + P(C) - P( R C) = 0.7 + 0.8 - 1 = 0.5

Therefore 0.5 is the required probability here.

g) P( R B) = P(R | B) P(B) = 0.896*0.375 = 0.336
Therefore 0.336 is the required probability here.

h) P(R C B)

= P( R B)P(C | B) = 0.336*0.8 = 0.2688
Therefore 0.2688 is the required probability here.