McMillan Math
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Theory 1

Repeated trials

When a single experiment type is repeated many times, and we assume each instance is independent of the others, we say it is a sequence of repeated trials or independent trials.

The probability of any sequence of outcomes is derived using independence together with the probabilities of outcomes of each trial.

A simple type of trial, called a Bernoulli trial, has two possible outcomes, 1 and 0, or success and failure, or and . A sequence of repeated Bernoulli trials is called a Bernoulli process.

  • Write sequences like for the outcomes of repeated trials of this type.
  • Independence implies
  • Write and , and because these are all outcomes (exclusive and exhaustive), we have . Then:
  • This gives a formula for the probability of any sequence of these trials.

A more complex trial may have three outcomes, , , and .

  • Write sequences like for the outcomes.
  • Label and and . We must have .
  • Independence implies
  • This gives a formula for the probability of any sequence of these trials.

Let stand for the sum of successes in some Bernoulli process. So, for example, “” stands for the event that the number of successes is exactly 3. The probabilities of events follow a binomial distribution.

Suppose a coin is biased with , and is ‘success’. Flip the coin 20 times. Then:

Each outcome with exactly 3 heads and 17 tails has probability . The number of such outcomes is the number of ways to choose 3 of the flips to be heads out of the 20 total flips.

The probability of at least 18 heads would then be:

With three possible outcomes, , , and , we can write sum variables like which counts the number of outcomes, and and similarly. The probabilities of events like follow a multinomial distribution.