probability practice question 1

 Consider a group of 600 students, of which 400 are female and 200 are MALE. Out of these, 100 have failed in the exam, and of these 100, 40 are MALE, F for female, x for failed, x' for not failed, and F' for MALE.


F: 400

F': 200

x: 100 

x': 500


create a table to show this


Gender       | Failed (x) | Not Failed (x')      | Total

---------------------------------------

Female  F        | 40            | 360                   | 400

---------------------------------------

Not Female F' | 60            | 140                    | 200

---------------------------------------

Total               | 100           | 500                    | 600


find probability of all intersections, union and conditionals


P(F): 400/600 = 2/3

P(F'): 200/600 = 1/3

P(x): 100/600 = 1/6

P(x'): 500/600 = 5/6


Intersections

P(F ∩ x): 40/600 = 0.067

P(F' ∩ x): 60/600 = 0.1

P(F ∩ x'): 360/600 = 0.6

P(F' ∩ x'): 140/600 = 0.233


Union

P(F ∪ x) = P(F) + P(x) - P(F ∩ x)

P(F ∪ x) = 400/600 + 100/600 - 40/600 = 0.7666


P(F' ∪ x): P(F') + P(x) - P(F' ∩ x) P(F' ∪ x) = 200/600 + 100/600 - 60/600 = 0.400


P(F ∪ x'): P(F) + P(x') - P(F ∩ x')

P(F ∪ x') = 400/600 + 500/600 - 360/600 = 0.900


P(F' ∪ x'): P(F') + P(x') - P(F' ∩ x')

P(F' ∪ x') = 200/600 + 500/600 - 140/600 = 0.9333


Conditionals

P(F | x): P(F ∩ x) / P(x)

P(F | x) = 40/100 = 0.4000


P(F' | x): P(F' ∩ x) / P(x)

P(F' | x) = 60/100 = 0.6000


P(F|x'): P(F ∩ x') / P(x') 

P(F|x') = 360/500 = 0.7200


P(F'|x'): P(F' ∩ x') / P(x')

P(F'|x') = 140/500 = 0.2800


P(x | F): P(F ∩ x) / P(F)

P(x | F) = 40/400 = 0.1000


P(x' | F): P(F ∩ x') / P(F)

P(x' | F) = 360/400 = 0.9000


P(x | F'): P(F' ∩ x) / P(F')

P(x | F') = 60/200 = 0.3000


P(x' | F'): P(F' ∩ x') / P(F')

P(x' | F') = 140/200 = 0.7000

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