4. Michael Phelps and Ryan Lochte are 2 of the US\'s top swimmers, and they both
ID: 3328891 • Letter: 4
Question
4. Michael Phelps and Ryan Lochte are 2 of the US's top swimmers, and they both will be swimming the 400 IM in the Olympics. Overall, Michael Phelps is known to have a 60% chance of winning the gold medal in the 400 IM. If Michael Phelps does not win the gold medal, Ryan Lochte has a 75% chance of winning the gold medal in the 400 IM. Overall, Ryan Lochte has a 30% chance of winning the Gold Medal in the 400 IM. Define: MP: the event Michael Phelps wins the Gold Medal in the 400 IM RL: the event Ryan Lochte wins the Gold Medal in the 400 IM a. Express the event "Michael Phelps wins the Gold Medal and Ryan Lochte does not win the Gold Medal" in terms of the events defined above b. What is the overall probability that neither Michael Phelps nor Ryan Lochte wins the Gold Medal (someone else wins it)? C. Given Ryan Lochte does not win the gold medal, what is the probability that Michael Phelps does win it? d. Are events MP and RL independent? How do you know? e. Are events MP and RL disjoint? How do you know?Explanation / Answer
P(MP) = 0.6 P(RL) = 0.3 P(RL|MPc) = 0.75
P(MPc) = 1 - 0.6 = 0.4
a. Michael Phelps wins the Gold Medal and Ryan Lochte does not win the Gold Medal
MP RLc
b. P(RL MPc) = P(RL|MPc) P(MPc)
= 0.75 * 0.4
= 0.3
P(RL MP) = P(RL) - P(RL MPc)
= 0.3 - 0.3 = 0 (Explains the fact that both cannot win the gold medal simultaneously)
P(RL U MP) = P(RL) + P(MP) - P(RL MP)
= 0.3 + 0.6 - 0
= 0.9
=> P(RLc MPc) = P(RL U MP)c = 1 - 0.9 = 0.1
c. P(RLc) = 1 - 0.3 = 0.7
P(MP RLc) = P(MP) - P(MP RL)
= 0.6 - 0
= 0.6
P(MP|RLc) = P(MP RLc) / P(RLc)
= 0.6 / 0.7
= 0.8571.
d. P(MP|RL) = P(MP RL) / P(RL)
= 0 / 0.3
= 0
P(MP) = 0.6
The events MP and RL are not independent since P(MP) and P(MP|RL) are not equal.
e. Since P(MP RL) = 0, MP and RL are disjoint.
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