# LearnBayes

Bayes: the science of evidence

## Probability of counts: the Binomial distribution

Suppose we are conducting an opinion poll for a political candidate, who we'll call Mr. Q. Mr. Q would like to know how many people support their candidacy over the candidate, Ms. K. For the sake of simplicity, suppose that everyone supports one or the other candidate, and we randomly sample people from the population to determine their opinion. We decide to sample 50 people; out of those 50 people, 36 support Mr Q. If, in reality, public opinion were evenly split, what was the probability that 36 people out of our sample would support Mr. Q?

## Conditional probability

At the heart of Bayesian statistics is the concept of conditioning; that is, quantifying uncertainty based on what you know. When we observe data, we wish to condition our beliefs based on those data. This article introduces conditional probability and ultimately Bayes rule.

## Bayes' theorem

Trisomy 21, or Down syndrome, is caused by having an extra copy of chromosome 21. Children with the disease typically show impairments in cognitive abilities and certain stereotypical physical features. Approximately 1 in 800 children have the disease. Prenatal tests for trisomy 21 exist, including the multiple-marker screening test performed in the second trimester of pregnancy.

The outcome of this screening test is either positive, meaning that the test indicates that the fetus has trisomy 21, or negative, indicating that it doesn't. Screening tests, however, are not perfect; sometimes a test will indicate the presence of the disease when the fetus does not have the disease (a "false positive"); likewise, the test can fail to detect the disease when the fetus actually does have the disease (a "false negative"). The false positive and false negative rates for the trisomy 21 screening are known, and are about 0.05 and 0.19, or 5% and 19%, respectively. A pregnant woman who obtains a positive screening result on the screening test must wonder: what is the probability that the baby will have trisomy 21?