A physical therapist of Russian football team knows that the team will play 40% of its matches on artificial turf, this season. Because of his vast experience, he knows that a football player’s chances of incurring a knee injury is 50% higher if he is playing on artificial turf instead of grass. If the player's chances of a knee injury is 0.42, what is the probability that a football player with knee injury, incurred the injury while playing on grass?
Explanation:
P(playing on artificial turf) = 0.4
∴ P(playing on grass) = 1 – P(playing on artificial turf) = 1 – 0.4 = 0.6
P(injury/artificial turf) = 0.42
Also, P(injury/artificial turf) = 1.5 × P(injury/grass)
∴ P(injury/grass) = 0.42/1.5 = 0.28
Now, P(injury) = P(injury/grass) × P(playing on grass) + P(injury/artificial turf) × P(playing on artificial turf)
= (0.28 × 0.6) + (0.42 × 0.4)
= 0.168 + 0.168 = 0.336
Also, P(injury/grass) × P(playing on grass) = P(grass/injury) × P(injury)
∴ P(grass/injury) = 0.168/0.336 = 0.5
Hence, option (c).
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