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Bernoulli Trials

Suppose we have a jar with 7 marbles, four of which are red and three are blue. A marble is drawn at random and we record whether it is red or blue.

Bernoulli Trials

Suppose we have a jar with \(7\) marbles, four of which are red and three are blue. A marble is drawn at random and we record whether it is red or blue. The probability \(p\) of getting a red marble is \(4/7\); and the probability of getting a blue is \(1-p=3/7\).

Now suppose the marble is put back in the jar, the marbles in the jar are stirred, and the experiment is repeated. Then the probability of getting a red marble on the second trial is again \(4/7\), and this pattern holds regardless of the number of times the experiment is repeated.

It is customary to call this situation a series of Bernoulli trials. More formally, we have an experiment with only two outcomes: success and failure. The probability of success is \(p\) and the probability of failure is \(1-p\). Most importantly, when the experiment is repeated, then the probability of success on any individual test is exactly \(p\).

We fix a positive integer \(n\) and consider the case that the experiment is repeated \(n\) times. The outcomes are then the binary strings of length \(n\) from the two-letter alphabet \(\{S,F\}\), for success and failure, respectively. If \(x\) is a string with \(i\) successes and \(n-i\) failures, then \(P(x)=\binom{n}{i}p ^i(1-p)^{n-i}\). Of course, in applications, success and failure may be replaced by: head/tails, up/down, good/bad, forwards/backwards, red/blue,

Example

When a die is rolled, let's say that we have a success if the result is a two or a five. Then the probability \(p\) of success is \(2/6=1/3\) and the probability of failure is \(2/3\). If the die is rolled ten times in succession, then the probability that we get exactly four successes is \(C(10,4)(1/3)^4 (2/3)^{6}\).

Example

A fair coin is tossed \(100\) times and the outcome (heads or tails) is recorded. Then the probability of getting heads \(40\) times and tails the other \(60\) times is \[\begin{aligned}\end{aligned}\]

Condensed — the full section is in Keller & Trotter, Applied Combinatorics.

Symbols used here

\binom{n}{k}

binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).

P(A),\ P(A \mid B)

probability, conditional probability
Chance of A; chance of A given that B happened.

imaginary unit
i² = −1.

factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

\int f(x)\,dx,\ \int_a^b

integral
Antiderivative (indefinite) or signed area from a to b (definite).

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\bar{x},\ \mu

sample mean, population mean
Average of the data; average of the whole population.

\sigma,\ s,\ \sigma^2

standard deviation, sample s.d., variance
Typical distance from the mean; its square.

E[X],\ \operatorname{Var}(X)

expected value, variance
Probability-weighted average of X; its spread.

N(\mu, \sigma^2),\ z

normal distribution, z-score
The bell curve with mean μ and variance σ²; (x − μ)/σ.

\mu(A),\ \sigma\text{-algebra}

measure of A
Size of a set; the family of sets that can be measured.

Questions people ask

What is the difference between probability and statistics?

Probability goes from a known model to what the data should look like; statistics goes from data back to the model. Probability theory is the deductive half.

What does the law of large numbers promise?

That the average of many independent samples converges to the expected value. It says nothing about any single trial.

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Parts of this page are adapted from Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed and re-explained here; errors are ours.