A random variable is a variable whose value is dependent on the outcome of a random event. This means the value is not known until the experiment is carried out, however the probabilities of all the outcomes can be modelled with a statistical distribution.
There are multiple ways of writing out a distribution:
The examples above are all the distribution of a fair, six-sided die. The probability of each outcome is the same, so it is called a discrete uniform distribution.
The sum of all probabilities in a distribution must equal 1
The Binomial Distribution
If you repeat an experiment multiples times (each time is known as a 'trial'), you can model the number of successful trials with the random variable, X.
A binomial distribution is used when:
There are a fixed number of trials, n
There are only two possible outcomes (success or fail)
The probability, p, of success is constant
All trials are independent
If the random variable X is distributed binomially with n number of trials and fixed probability of success p, it is noted as:
Generally, you use a calculator to calculate binomial problems, but you can also use the probability mass function:
There are two forms of the binomial distribution: exact and cumulative:
Exact Binomial Problems
This is when a question asks you to find the probability of there being a specific number of successes out of the number of trials, n.