1 [15 pts]. Krishna\'s Bakery has two dedicated customers (Matt and Shawn) that,

Question

1 [15 pts]. Krishna's Bakery has two dedicated customers (Matt and Shawn) that, every morning, buy an assortment of a dozen baked goods each. Each pastry Matt buys is a cupcake with probabilityand a donut with probability , while each pastry Shawn buys is a donut with probability and a cupcake with probability . Matt and Shawn are equally likely to arrive at the register first, and once a customer reaches the register, they chose one item at a time to buy with the aforementioned probabilities, until they have purchased their dozen (whoever comes second has to wait for the other to finish before ordering) (a) Show that the probability of a donut being the first baked good bought on any given day is (b) If the first two baked goods bought are cupcakes, what is the probability the third baked good bought is a donut? (c) If the first two baked goods bought are donuts, what is the probability that Matt was the first to arrive at the register?

Explanation / Answer

Here, we are given that:

P( cupcake | Matt ) = (2/3)
P( donut | Matt ) = (1/3)

P( cupcake | Shawn ) = (1/3)
P( donut | Shawn ) = (2/3)

P( Matt ) = P( Shawn ) = 0.5

a) According to the law of total probability, we get here:

P( donut ) = P( donut | Matt) P( Matt) + P( donut | Shawn )P(Shawn )

P( donut ) = (1/3)*0.5 + (2/3)*0.5 = 0.5

Therefore 0.5 is the required probability here. Hence proved.

b) P( first two ordered are cupcakes )

= P( first two ordered are cupcakes | Matt )P( Matt ) + P( first two ordered are cupcakes | Shawn )P( Shawn )

= (2/3)2*0.5 + (1/3)2*0.5

= 5/18

Now given that the first two products are cupcakes, probability that third one is a donut is computed here as:

= [ (1/3)*P( Matt | first two ordered are cupcakes ) + (2/3)*P( Shawn | first two ordered are cupcakes ) ] / P( first two ordered are cupcakes )

= [ (1/3)*(4/18) + (2/3)*(1/18) ] / (5/18)

= 0.4

Therefore 0.4 is the required probability here.

c) Given that first two ordered are donuts,

P( first two ordered are donuts ) = (1/3)2*0.5 + (2/3)2*0.5 = (5/18)

P( Matt | first two ordered are donuts ) = (1/18) / (5/18) = 1/5 = 0.2

Therefore 0.2 is the required probability here.

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