# Thread: the probability that at least one of them will have a thyroid deficiency

1. ## 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?

2. Originally Posted by Yan
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?
Tell us the probability that none of the twelve has the condition.

3. 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!. $\displaystyle P(B) = 0.9^{12}$.

$\displaystyle P(B) = 1 - P(B)$

$\displaystyle P(B) = 1 - 0.9^{12}$

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### medical records show that one out of 10 persons

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