Independent Events and Bernoulli Trials An independent event is an event whose outcome is independent of the outcomes of other events. This means that th...
An independent event is an event whose outcome is independent of the outcomes of other events. This means that the outcome of one event does not affect the outcome of another event, regardless of their relative positions or timings.
For example, if you roll a die and then spin it, the outcome of the roll will be independent of the outcome of the spin. Similarly, the outcome of one toss of a coin will be independent of the outcome of a different toss.
A Bernoulli trial is a special type of independent event in which there are only two possible outcomes: success or failure. The probability of success is constant for each trial, and it remains the same regardless of the number of trials or the outcome of previous trials.
Examples of independent events include:
Rolling a different number on a dice each time
A coin landing on heads or tails
Two people randomly choosing the same option on a multiple-choice question
Examples of Bernoulli trials include:
Rolling a die to determine the number of dots showing
A coin landing on heads or tails
A patient receiving a medication or a treatment
The probability of success in a Bernoulli trial can be calculated using the following formula:
P(success) = p
where:
P(success) is the probability of the event occurring
p is the probability of the event happening on each trial
The expected number of trials until the first success is given by the following formula:
E(X) = n*p
where:
E(X) is the expected number of trials before the first success
n is the total number of trials
p is the probability of success on each trial
In summary, independent events are events whose outcomes are independent of the outcomes of other events, and Bernoulli trials are a special type of independent event in which there are only two possible outcomes