Medical records show that one out of 10 persons in a certain town has a thyroid deficiency. If 12 persons in this town are randomly chosen and tested, what is the probability that at least one of them will have a thyroid deficiency?
Medical records show that one out of 10 persons in a certain town has a thyroid deficiency. If 12 persons in this town are randomly chosen and tested, what is the probability that at least one of them will have a thyroid deficiency?
Let $\displaystyle P(A) $ be the probability of at least 1 person from 12 having the deficiency.
Let $\displaystyle P(B) $ be the probability of nobody from 12 having the deficiency.
We can conclude that $\displaystyle P(A) = 1 - P(B) $
Now we just need $\displaystyle P(B) $
Now. If you have 12 people, and each of them has a 0.1 chance of having the deficiency. Then you can say that 12 people have a 0.9 chance of NOT having the deficiency. So what's the probability that ALL of will NOT have the deficiency? Well we know that if two events have 2 probabilities of occuring, then the probability of both occuring is the product of their probabilities. This works the same, although instead of the product of 2 probabilities, we have the product of 12!. $\displaystyle P(B) = 0.9^{12} $.
$\displaystyle P(B) = 1 - P(B) $
$\displaystyle P(B) = 1 - 0.9^{12} $