Independent Events: An independent event is an event whose occurrence does not depend on the occurrence of other events. This means that the probability of...
Independent Events: An independent event is an event whose occurrence does not depend on the occurrence of other events. This means that the probability of one event occurring does not change if another event occurs, regardless of the timing between the events. For example, if you roll a die and then flip a coin, the outcome of the coin flip is independent of the outcome of the die roll.
Bernoulli Trials: A Bernoulli trial is a discrete experiment with only two possible outcomes, often called success and failure. The probability of success must be equal to the probability of failure. A simple example of a Bernoulli trial is rolling a coin and recording whether it lands heads or tails.
Probability: In a Bernoulli trial with n trials and only two possible outcomes, the probability of an event occurring is given by the formula:
where:
n is the total number of trials
n_E is the number of successes
n is the total number of trials
Independent Events and Bernoulli Trials: The independence of events means that the probability of both events occurring is equal to the product of the individual probabilities:
This property allows us to combine multiple Bernoulli trials into a single Bernoulli trial with the same probability of success.
Examples:
Rolling a 6 on a standard dice is an independent event from the outcome of flipping a coin.
Getting a heads or tails outcome in a single coin flip is a Bernoulli trial.
Rolling all the balls in a bag without looking at them is an independent event, and the probability of each ball being picked is equal.
Applications of Bernoulli Trials: Bernoulli trials have various applications in probability and statistics, including modeling random experiments, calculating probabilities of specific outcomes, and testing hypotheses