2) Consider a probability space where outcomes are not equally likely. For examp

Question

2) Consider a probability space where outcomes are not equally likely. For example, in the arcade game called Skee Ball, players roll a ball up a ramp and it lands in different areas, which are worth different points ranging from 10 to 50. Since the areas are of different sizes, if a ball is rolled randomly up the ramp, the probabilities that the ball lands in different areas are not equal. Assume that the following probablities are associated with the different areas: Value 10 points 20 points 30 points 40 points 50 points Probabili 0.5 0.25 0.15 0.08 0.02 If a player rolls 3 balls at random: a) What is the probability of a total score of at least 100 points? b) Given that the first ball scores 30 points, what is the probability of a total score of at least 100 points? Are the events "the first ball scores 30% and "the total score is at least 100% independent? Why or why not? c) What is the expected total score? What is the variance of the score on one roll?

Explanation / Answer

a)

So total probability is 0.02805

b)

If 30 is the first ball then,

So probability is 0.0075

c)

Ball 1 Ball 2 Ball3 Total P(B1) P(B2) P(B3) P(Total >=100) 10 40 50 100 0.5 0.08 0.02 0.0008 10 50 40 100 0.5 0.02 0.08 0.0008 10 50 50 110 0.5 0.02 0.02 0.0002 20 30 50 100 0.25 0.15 0.02 0.00075 20 50 30 100 0.25 0.02 0.15 0.00075 20 40 50 110 0.25 0.08 0.02 0.0004 20 50 40 110 0.25 0.02 0.08 0.0004 20 40 40 100 0.25 0.08 0.08 0.0016 20 50 50 120 0.25 0.02 0.02 0.0001 30 20 50 100 0.15 0.25 0.02 0.00075 30 30 40 100 0.15 0.15 0.08 0.0018 30 30 50 110 0.15 0.15 0.02 0.00045 30 40 30 100 0.15 0.08 0.15 0.0018 30 40 40 110 0.15 0.08 0.08 0.00096 30 40 50 120 0.15 0.08 0.02 0.00024 30 50 20 100 0.15 0.02 0.25 0.00075 30 50 30 110 0.15 0.02 0.15 0.00045 30 50 40 120 0.15 0.02 0.08 0.00024 30 50 50 130 0.15 0.02 0.02 0.00006 40 10 50 100 0.08 0.5 0.02 0.0008 40 20 40 100 0.08 0.25 0.08 0.0016 40 20 50 110 0.08 0.25 0.02 0.0004 40 30 30 100 0.08 0.15 0.15 0.0018 40 30 40 110 0.08 0.15 0.08 0.00096 40 30 50 120 0.08 0.15 0.02 0.00024 40 40 20 100 0.08 0.08 0.25 0.0016 40 40 30 110 0.08 0.08 0.15 0.00096 40 40 40 120 0.08 0.08 0.08 0.000512 40 40 50 130 0.08 0.08 0.02 0.000128 50 10 40 100 0.02 0.5 0.08 0.0008 50 10 50 110 0.02 0.5 0.02 0.0002 50 20 30 100 0.02 0.25 0.15 0.00075 50 20 40 110 0.02 0.25 0.08 0.0004 50 20 50 120 0.02 0.25 0.02 0.0001 50 30 20 100 0.02 0.15 0.25 0.00075 50 30 30 110 0.02 0.15 0.15 0.00045 50 30 40 120 0.02 0.15 0.08 0.00024 50 30 50 130 0.02 0.15 0.02 0.00006 50 40 10 100 0.02 0.08 0.5 0.0008 50 40 20 110 0.02 0.08 0.25 0.0004 50 40 30 120 0.02 0.08 0.15 0.00024 50 40 40 130 0.02 0.08 0.08 0.000128 50 40 50 140 0.02 0.08 0.02 0.000032 50 50 10 110 0.02 0.02 0.5 0.0002 50 50 20 120 0.02 0.02 0.25 0.0001 50 50 30 130 0.02 0.02 0.15 0.00006 50 50 40 140 0.02 0.02 0.08 0.000032 50 50 50 150 0.02 0.02 0.02 0.000008 0.02805

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