A swimmer is trying to qualify for nationals in the backstroke and the freestyle

Question

A swimmer is trying to qualify for nationals in the backstroke and the freestyle. The swimmer believes that the probability of qualifying in the backstroke is 0.8, the probability of qualifying in the freestyle is 0.6, and the probability of qualifying on both is 0.5. Find the following probabilities: Please show how you came to the answers

a) The student does not qualify in the backstroke.  

b) The student qualifies in the backstroke, but not the freestyle.   

c) The student qualifies on the freestyle, but not the backstroke.

d) The student does not qualify on either event.  

Explanation / Answer

a) P(Does not qualify in backstroke)

= 1 - P(Qualifies in backstroke)

= 1 - 0.8

= 0.2

b) P(Qualifies in backstroke but not in freestyle)

= P(Qualifies in backstroke) - P(Qualifies in both)

= 0.8 - 0.5

= 0.3

c) P(Qualifies in freestyle but not in backstroke)

= P(Qualifies in freestyle) - P(Qualifies in both)

= 0.6 - 0.5

= 0.1

d) P(Does not qualifies in either event)

= 1 - P(Qualifies in atleast one of the events)

= 1 - [P(Qualifies in bacstroke) + P(Qualifies in freestyle) - P(Both)]

= 1 - [0.8 + 0.6 - 0.5]

= 1 - 0.9

= 0.1

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