X is taking up subjects – Mathematics, Physics and Chemistry in the examination. His probabilities of getting grade A in these subjects are 0.2, 0.3 and 0.5 respectively. Find the probability that he gets
i. Grade A in all subjects
ii. Grade A in no subjects
iii. Grade A in two subjects
Given:
⇒ P(MA) = P(getting A in mathematics)
⇒ P(MA) = 0.2
⇒ P(MN) = P(not getting A in mathematics)
⇒ P(MN) = 1-0.2
⇒ P(MN) = 0.8
⇒ P(PA) = P(getting A in physics)
⇒ P(PA) = 0.3
⇒ P(PN) = P(not getting A in physics)
⇒ P(PN) = 1-0.7
⇒ P(PN) = 0.3
⇒ P(CA) = P(getting A in Chemistry)
⇒ P(CA) = 0.5
⇒ P(CN) = P(not getting A in chemistry)
⇒ P(CN) = 1-0.5
⇒ P(CN) = 0.5
We need to find the probability that:
i. X gets A in all subjects
ii. X gets A in no subjects
iii. X gets A in two subjects
⇒ P(Xall) = P(getting A in all subjects)
Since getting A in different subjects is an independent event, their probabilities multiply each other
⇒ P(Xall) = (P(MA)P(PA)P(CA))
⇒ P(Xall) = 0.2×0.3×0.5
⇒ P(Xall) = 0.03
⇒ P(Xnone) = P(getting A in no subjects)
Since getting A in different subjects is an independent event, their probabilities multiply each other
⇒ P(Xnone) = (P(MN)P(PN)P(CN))
⇒ P(Xnone) = 0.8×0.7×0.5
⇒ P(Xnone) = 0.28
⇒ P(Xtwo) = P(getting A in any two subjects)
Since getting A in different subjects is an independent event, their probabilities multiply each other
⇒ P(Xtwo) = (P(MA)P(PA)P(CN)) + (P(MA)P(PN)P(CA)) + (P(MN)P(PA)P(CA))
⇒ P(Xtwo) = (0.2×0.3×0.5) + (0.2×0.7×0.5) + (0.8×0.3×0.5)
⇒ P(Xtwo) = 0.03 + 0.07 + 0.12
⇒ P(Xtwo) = 0.22
∴ The required probabilities are 0.03, 0.28, 0.22.