You have been stranded on a deserted island for 7 years, when one day a storm bl

Question

You have been stranded on a deserted island for 7 years, when one day a storm blows a shipwreck onto the island. In the wreck you see a waterlogged case containing a flare gun with 5 flares. They are wet however, and retrieving the case would be very dangerous You estimate that there is only a probability of 0.2 for any one of the flares to go off. Use this information to answer the following questions. 8. A) What is the probability that fewer than two flares go off? B) What is the probability that between two and four flares, inclusive, go off? C) What is the probability that all five flares go off? You believe if fewer than two flares go off that you will never be rescued, if between two and four flares go off it will take you years to be rescued, and if all five flares go off that you will be rescued in a matter of months. You assign units of happiness (Utils) to these outcomes as shown in the table below. Suppose that if you do not risk your life to get the case, that you will have 1000 Utils of happiness over the rest of your life Time Never Years Months Util -2000 7500 75000 D) What is the expected value of Utls you will achieve if you obtain the flare gun? E) Will you attempt to retrieve the case or not?

Explanation / Answer

Probability of a flare going off =0.2

Probability of a flare not going off =1-0.2 =0.8

(a) Probability that fewer than 2 flares go off = P(one flare goes off) + P(no flare goes off) = C [5,1] x 0.2^1 x 0.8^4 + 0.8^5 = 0.4096 + 0.3276 = 0.7373

(b) P(between 2 and 4 flares go off) = P(2 flares go off) + P(3 flares go off) + P(4 flares go off) = C[5,2] x 0.2^2 x 0.8^3 + C[5,3] x 0.2 ^3 x 0.8^2 + C[5,4] x 0.2^4 x 0.8^1 = 0.2048 + 0.0512 + 0.0064 = 0.2624

(c) P(All five flares go off) = 0.2^5 = 0.00032

(d) Expected value of Utils = 1000 -2000 x 0.7373 + 7500 x 0.2624 + 75000 x 0.00032 = 1517.4

(e) Since the expected value is positive, one will attempt to retrive the case

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