Question

8. The Probability Calculus - Bayes's Theorem

Bayes's Theorem is used to calculate the conditional probability of two or more events that are mutually exclusive and jointly exhaustive. An event's conditional probability is the probability of the event happening given that another event has already occurred. The probability of event A given event B is expressed as P(A given B). If two events are mutually exclusive and jointly exhaustive, then one and only one of the two events must occur.

Your textbook restricts the number of mutually exclusive and jointly exhaustive events to two. When two events (A11 and A22) are mutually exclusive and jointly exhaustive, Bayes's Theorem is expressed as follows:

  

Consider the following scenario. Determine the fractional probabilities for the following events: P(A11), P(A22), P(B given A11), P(B given A22), as indicated next. Reduce the fractional answers to the lowest whole numbers and type your responses into the spaces provided. Then calculate the decimal probability of event P(A11 given B) using Bayes's Theorem. Type your numeric answer, written out to four decimal places, into the last space indicated.

Suppose you are given seven blue urns and three green urns, in a dark room so that you cannot tell the difference between the urns. Each blue urn contains one red ball and nine yellow balls. Each green urn contains four red balls and one yellow ball. You draw a ball at random from one of the urns.

Let event A11 be drawing a ball from a blue urn. What is the probability of event A11 expressed as a fraction? P(A11) =  /

Let event A22 be drawing a ball from a green urn. What is the probability of event A22 expressed as a fraction? P(A22) =  /

Let event B be drawing a red ball. Assuming the urn is blue, what is the probability of drawing a red ball from the blue urn, expressed as a fraction? P(B given A11) =  /

Let event B be drawing a red ball. Assuming the urn is green, what is the probability of drawing a red ball from the green urn, expressed as a fraction? P(B given A22) =  /

Assuming you draw a red ball from one of the urns, what is the (decimal) probability that the urn was blue according to Bayes's Theorem? P(A11 given B) =

Answer 1

There are 7 ways of selecting blue urns out of 10, therefore (A11) = 7/10

There are 3 ways of selecting green urns out of 10, therefore, P(A22) = 3/10

Since there is 1 way of selecting red ball from blue urn,  P(B given A11) = (1/10)

Since there are 4 ways of selecting red ball from green urn, P(B given A22) = (4/5)

P(A11 given B) = P(B given A11)*P(A11) / P(B)

P(B) = P(B given A11).*P(A11) + P(B given A22)* P(A22)

= (1/10)*(7/10) + (4/5)*(3/10)

= 31/100

P(A11 given B) = (7/100) / (31/100) = 7/31