Jake woke up late in the morning on the day that he has to go to school to take
ID: 3153675 • Letter: J
Question
Jake woke up late in the morning on the day that he has to go to school to take an important test. He can either take the shuttle bus which is usually running late 20% of the time or ride his unreliable motorcycle which breaks down 40% of the time. He decides to toss a fair coin to make his choice.
If Jake, in fact, gets to the test on time, what is the probability that he took the bus?
If Jake, in fact, gets to the test on time, what is the probability that he rode his bike?
If Jake is late to the test, what is the probability that he rode his bike?
If Jake is late to the test, what is the probability that he took the bus?
Explanation / Answer
Let
B = bus
M = motorcycle
O = on time
If Jake, in fact, gets to the test on time, what is the probability that he took the bus?
Hence, by Bayes' Rule,
P(O) = P(B) P(O|B) + P(M) P(O|M) = (1/2)*(1-0.40) + (1/2)*(1-0.20) = 0.7
Thus,
P(B|O) = P(B) P(O|B)/P(O) = (1/2)*(1-0.40)/0.7 = 0.428571429 [ANSWR]
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If Jake, in fact, gets to the test on time, what is the probability that he rode his bike?
Also,
P(M|O) = P(M) P(O|M)/P(O) = (1/2)*(1-0.20)/0.7 = 0.571428571 [ANSWER]
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If Jake is late to the test, what is the probability that he rode his bike?
As
P(O') = 1 - P(O) = 1 - 0.7 = 0.3
Hence,
P(M|O') = P(M) P(O'|M)/P(O') = (1/2)*0.40/0.3 = 0.666666667 [ANSWER]
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If Jake is late to the test, what is the probability that he took the bus?
P(B|O') = 1 - P(M|O') = 1 - 0.6666667 = 0.333333333 [ANSWER]
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