Bernoulli Trial is an experiment in which the outcome is either a SUCCESS or a FAILURE. SUCCESS means that you are getting the result that you’re counting, but it does not necessarily mean the traditional meaning of triumph or prosperity.

The probability of each outcome is independent of the results of the previous trial. The probability of each possible outcome is the same for each trial. One example of a Bernoulli trial is the coin-tossing experiment, which results in either head or tail.

The Bernoulli trials, named by Jacques (Jacob) Bernoulli, born in Switzerland, is one of the simplest yet most important random processes in probability.

A sequence of Bernoulli Trials satisfies the following assumptions:

- Each trial has two possible outcomes, called SUCCESS or FAILURE.
- The trials are independent, meaning that they do not influence each other.
- On each trial, the probability of success is \( x \), and the probability of failure is \( 1-x \).

## Example Cases

Determine which can be defined as a Bernoulli trial.

### Case 1

It rolls a die and records the number that comes up.

No, because the outcome is \( 1, \ 2, \ 3, \ 4, \ 5 \) or \( 6 \).

### Case 2

It rolls a die five times and records the number of \(3\)s that come up.

Yes, because the outcome is either \( 3 \) or not \( 3 \).

### Case 3

Spinning a spinner numbered \(1\) to \(8\) and recording the obtained number.

No, because the outcome is more than two.

### Case 4

Tossing a coin seven times and recording the number of heads obtained.

Yes, the outcome is either a head or a tail.

### Case 5

Drawing a card five times from a fair deck *without* replacement and recording the number of red cards.

No, the outcome is dependent because each trial’s probabilities are different.

### Case 6

Drawing a card five times from a fair deck *with* replacement and recording the number of black cards.

Yes, the outcome is independent and is either red or black.

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