Answer to Question #443 in Statistics and Probability for Elina

Question #443
It is known that in average there is 1% of left-handers. Estimate the probability that at least four left-handers will be among: a) 200 people; b) 10000 people.
Expert's answer

If left-handers make up 1%, then among 400 people there will be 4 left-handers with the probability 1.

a) You can select 200 out of 400 in (C400)^200 ways. Consider the event that there are 4 left-handers and 196 right-handers among 200 people. 4 out of 4 left-handers can be got in (C4)^4=1 way. 196 right-handers can be got out of 396 in (C396)^196 ways. Thus, the number of options of selecting people in which among 200 people there will be 4 left-handers is: 1*(C396)^196 = (C396)^196. The probability is: P = ((C396)^196)/((C400)^200) = (396!*200!*200!)/(400!*200!*196!) ≈ 0.062.

b) Left-handers make up 1%, so among 10000 people there would be 100 left-handers, then at least 4 left-handers will be with the probability 1.

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