Question

1. Emily, Car, Stock Market, Sweepstakes, Vacation and Bayes.

Emily is taking Bayesian Analysis course. She believes she will get an A with probability 0.6, a B with probability 0.3, and a C or less with probability 0.1. At the end of semester, she will get a car as a present form her uncle depending on her class performance. For getting an A in the course Emily will get a car with probability 0.8, for B with probability 0.5, and for anything less than B, she will get a car with probability of 0.2. These are the probabilities if the market is bullish. If the market is bearish, the uncle is less likely to make expensive presents, and the above probabilities are 0.5, 0.3, and 0.1, respectively. The probabilities of bullish and bearish market are equal, 0.5 each. If Emily gets a car, she would travel to Redington Shores with probability 0.7, or stay on campus with probability 0.3. If she does not get a car, these two probabilities are 0.2 and 0.8, respectively. Independently, Emily may be a lucky winner of a sweepstake lottery for a free air ticket and vacation in hotel Sol at Redington Shores. The chance to win the sweepstake is 0.001, but if Emily wins, she will go to vacation with probability of 0.99, irrespective of what happened with the car.

After the semester was over you learned that Emily is at Redington Shores.

(a) What is the probability that she won the sweepstakes?

Answer 1

1.(a) Suppose,

R denotes going to Redington Shores.

S denotes wining a sweepstakes.

X denotes the situation of getting a car or not.

By Bayes' theorem,

;[ P(R|X)*P(X) = P(RX)/P(X)*P(X) =P(RX) ]

P(R|S)=0.99, P(S)=0.001

P(Car) = P(Car | Market is Bullish)*P(Market is Bullish) + P(Car | Market is Bearish)*P(Market is Bearish)

= [P(Car | Market is Bullish, Obtained Grade A)*P(Grade A)+P(Car | Market is Bullish, Obtained Grade B)*P(Grade B)    +P(Car | Market is Bullish, Obtained Grade C)*P(Grade C)] * P(Market is Bullish)

+ [P(Car | Market is Bearish, Obtained Grade a)*P(Grade A)+P(Car | Market is Bearish, Obtained Grade B)*P(Grade B) +P(Car | Market is Bearish, Obtained Grade C)*P(Grade C)] * P(Market is Bearish)

   = (0.8*0.6+0.5*0.3+0.2*0.1)*0.5+(0.5*0.6+0.3*0.3+0.1*0.1)*0.5 = 0.525

P(Not getting a car) = 1-0.525 =0.475

P(RX) = P(R | Car)*P(Car)+P(R | Not getting a car)*P(Not getting a car) = 0.7*0.525+0.2*0.475 = 0.4625

So, P(S|R) = 0.99*0.001 / (0.99*0.001+0.4625) = 0.0021