Chapter 1: Bernoulli Trial
A Bernoulli trial is a random experiment where there are only two possible outcomes: success or failure.
The probability of success is denoted by \(p\), and the probability of failure is \(1 - p\).
To make this concept intuitive, imagine a game where you roll a fair six-sided die.
You win a prize if the number is a multiple of 3, which includes 3 and 6.
There are two favorable outcomes out of six:
\[
p = \frac{2}{6} = \frac{1}{3}
\]
Let \(X\) be a random variable that represents the outcome of a Bernoulli trial:
\(X = 1\) if the outcome is a success
\(X = 0\) if the outcome is a failure
The probability mass function (PMF) of \(X\) is:
\[ P(X = x) = p^x (1 - p)^{1 - x}, \quad x \in \{0, 1\} \]
Example Calculation:
Let \(p = \frac{1}{3}\)
- \(P(X = 1) = \left(\frac{1}{3}\right)^1 \cdot \left(\frac{2}{3}\right)^0 = \frac{1}{3}\)
- \(P(X = 0) = \left(\frac{1}{3}\right)^0 \cdot \left(\frac{2}{3}\right)^1 = \frac{2}{3}\)
In R:
## success failure
## 0.3333333 0.6666667A Bernoulli trial is the foundation for many other distributions in probability theory, including the binomial distribution, which models multiple independent Bernoulli trials.
In the context of ddPCR, each droplet behaves like a Bernoulli trial: it either contains one or more copies of the target DNA (success), or it contains none (failure).