Terms you'll find helpful in completing today's challenge are outlined below.

Random Variable

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.

Binomial Experiment

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 ().
  
  

Bernoulli Random Variable and Distribution

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:

Binomial Distribution

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.

Cumulative Probability

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 :

Example

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: