Terms you'll find helpful in completing today's challenge are outlined below.
A random variable, , is the real-valued function in which there is an event for each interval where . You can think of it as the set of probabilities for the possible outcomes of a sample space. For example, if you consider the possible sums for the values rolled by four-sided dice:
Note: When we roll two dice, the value rolled by each die is independent of the other.
A binomial experiment (or Bernoulli trial) is a statistical experiment that has the following properties:
- The experiment consists of repeated trials.
- The trials are independent.
- The outcome of each trial is either success () or failure ().
The sample space of a binomial experiment only contains two points, and . We define a Bernoulli random variable to be the random variable defined by and . If we consider the probability of success to be and the probability of failure to be (where ), then the probability mass function (PMF) of is:
We can also express this as:
We define a binomial process to be a binomial experiment meeting the following conditions:
- The number of successes is .
- The total number of trials is .
- The probability of success of trial is .
- The probability of failure of trial , where .
- is the binomial probability, meaning the probability of having exactly successes out of trials.
The binomial random variable is the number of successes, , out of trials.
The binomial distribution is the probability distribution for the binomial random variable, given by the following probability mass function:
Note: Recall that . For further review, see the Combinations and Permutations Tutorial.
We consider the distribution function for some real-valued random variable, , to be . Because this is a non-decreasing function that accumulates all the probabilities for the values of up to (and including) , we call it the cumulative distribution function (CDF) of . As the CDF expresses a cumulative range of values, we can use the following formula to find the cumulative probabilities for all :
A fair coin is tossed times. Find the following probabilities:
- Getting heads.
- Getting at least heads.
- Getting at most heads.
For this experiment, , , and . The respective probabilities for the above three events are as follows:
- The probability of getting heads is:
- The probability of getting at least heads is:
- The probability of getting at most heads is: