conditional probability question - Sanjay the kindergartener with chicken pox

Question

Suppose the following facts to be true:

-- The probability of a random kindergartener having chicken pox at any given time is 2%.

-- Among kindergarteners who have chicken pox, 75% have red spots.

-- Among kindergarteners who do not have chicken pox, 1% have red spots.

Given that Sanjay, a kindergartener, has red spots, what is the probability that Sanjay has chicken pox?

This is my reasoning:

We know for sure that Sanjay has red spots. 75% of children who have chicken pox have red spots. That means 25% of children who have chicken pox DO NOT have red spots. The probability of Sanjay having chicken pox is 2%. Would it be 75% of 2%? I'm not very sure. This question is a bit confusing.

Answer 1

This is a standard conditional probability question. By definition:

$$ P(\text{chicken pox} \mid \text{spots}) = \frac{P(\text{chicken pox} \cap \text{spots})}{P(\text{spots})} $$

So we now have to find two probabilities:

1. probability of a random person having chicken pox and spots
2. probability of a random person having spots

The first probability is given in the question, $2\%$ of people have chicken pox and of those, $75\%$ have red spots.

The second probability is marginally more difficult. $2\%$ of people have chicken pox and $75\%$ of those have red spots. We must also consider that $100\% - 2\%=98\%$ of people do not have chicken pox, and $1\%$ of those have red spots. Note that these events $\text{chicken pox} \cap \text{spots}$ and $\text{no chicken pox} \cap \text{spots}$ are disjoint (mutually exclusive) and therefore the probability of either happening is simply the sum of their probabilities.

Answer 2

Formally, you can apply Bayes's Theorem.

Informally, notice that $1.5$% of kindergartners have chicken pox and have red spots, while $0.98$% ($1$% of the other $98$%)have red spots but do not have chicken pox. That is, out of $10000$ kindergartners, you would on average find $248$ with red spots, of which $150$ had chicken pox and $98$ did not.

We know Sanjay is among the $248$, and must estimate the probability that he is in the first subset of those children.