What are the possible values of the random variable for the number of heads if a coin was tossed?

home / probability and statistics / inferential statistics / random variable

A random variable in probability and statistics is described as a variable in which the values of the variable are dependent on the outcomes of a random phenomenon or experiment. Random variables can take on a set of different possible values, each of which has a certain probability of occuring. Random variables are different from the type of variable used in algebra, and can be thought of as a way to map the potential outcomes of a random phenomenon.

A random event happens by chance; it is unpredictable. For example, whether a tossed coin lands on "heads" or "tails" is random. It's also possible for the penny to land on its side, but these events are not considered. Random variables allow us to quantify the outcomes of tossing a coin by assigning values to the outcomes. We only consider 2 outcomes of the coin toss, heads or tails, so we can use the numbers 1 and 0 to represent these outcomes and define a random variable, X, as:

The random variable X can take on the value of 1 if the coin lands on tails, or 0 if the coin lands on heads, with a probability of 50% of either occuring in a given toss of the coin. The values that X (1 and 0 in this case) can take on are referred to as the sample space, which can be denoted as {0, 1}. If we then wanted to write the probability of the coin landing on heads, we could write P(X = 0). For tails it would be P(X = 1). Using random variables allows us to efficiently write expressions and perform calculations by quantifying random events, rather than having to write out something like "the probability of a coin landing on heads when flipped one time." A coin flip is a relatively simple example of a random event. As the events we want to study get more complex, the use of random variables becomes even more useful.

Discrete vs continuous random variables

A discrete random variable is a variable that can only take on certain exact values; they can either take on a finite number of distinct values, or a countably infinite set of values, like the integers. Another example of a discrete random variable is the possible values from the roll of a 6-sided die (1, 2, 3, 4, 5, 6).

In contrast, a continuous random variable takes on all values in a given interval. For example, given the interval [1, 5], a continuous random variable includes all real numbers (1.001, 2.889000015, 4) not just integer values. There are an infinite number of possibilities when considering continuous random variables. Examples of continuous random variables include human height and weight. While there is a certain range within which these measurements fall, within that range, there are an infinite number of possible height and weight measurements.

Get the answer to your homework problem.

Try Numerade free for 7 days

We don’t have your requested question, but here is a suggested video that might help.

A coin is to be tossed until a head appears twice in a row. What is the sample space for this experiment? If the coin is fair, what is the probability that it will be tossed exactly four times?

Answer:

There is a 50% chance the coin will face heads if it is a double-sided coin.

Explanation:

A coin is double-sided, so the random variable that it might point heads is 50%. But if it was a triple sided coin, its random variable will be 33.33%. Take the other side, you'll be left with 66.7%, and you'll get a 33.33% chance to get heads, and a 33.34% chance to get tails.

A Random Variable is a set of possible values from a random experiment.

Let's give them the values Heads=0 and Tails=1 and we have a Random Variable "X":

In short:

X = {0, 1}

Note: We could choose Heads=100 and Tails=150 or other values if we want! It is our choice.

So:

• We have an experiment (such as tossing a coin)
• We give values to each event
• The set of values is a Random Variable

Not Like an Algebra Variable

In Algebra a variable, like x, is an unknown value:

In this case we can find that x=4

But a Random Variable is different ...

A Random Variable has a whole set of values ...

... and it could take on any of those values, randomly.

X could be 0, 1, 2, or 3 randomly.

And they might each have a different probability.

Capital Letters

We use a capital letter, like X or Y, to avoid confusion with the Algebra type of variable.

Sample Space

A Random Variable's set of values is the Sample Space.

Example: Throw a die once

Random Variable X = "The score shown on the top face".

X could be 1, 2, 3, 4, 5 or 6

So the Sample Space is {1, 2, 3, 4, 5, 6}

Probability

We can show the probability of any one value using this style:

P(X = value) = probability of that value

X = {1, 2, 3, 4, 5, 6}

In this case they are all equally likely, so the probability of any one is 1/6

• P(X = 1) = 1/6
• P(X = 2) = 1/6
• P(X = 3) = 1/6
• P(X = 4) = 1/6
• P(X = 5) = 1/6
• P(X = 6) = 1/6

Note that the sum of the probabilities = 1, as it should be.

X = "The number of Heads" is the Random Variable.

In this case, there could be 0 Heads (if all the coins land Tails up), 1 Head, 2 Heads or 3 Heads.

So the Sample Space = {0, 1, 2, 3}

But this time the outcomes are NOT all equally likely.

The three coins can land in eight possible ways:

		X =   "Number of Heads"
HHH			3
HHT			2
HTH			2
HTT			1
THH			2
THT			1
TTH			1
TTT			0

Looking at the table we see just 1 case of Three Heads, but 3 cases of Two Heads, 3 cases of One Head, and 1 case of Zero Heads. So:

• P(X = 3) = 1/8
• P(X = 2) = 3/8
• P(X = 1) = 3/8
• P(X = 0) = 1/8

Example: Two dice are tossed.

The Random Variable is X = "The sum of the scores on the two dice".

Let's make a table of all possible values:

There are 6 × 6 = 36 possible outcomes, and the Sample Space (which is the sum of the scores on the two dice) is {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Let's count how often each value occurs, and work out the probabilities:

• 2 occurs just once, so P(X = 2) = 1/36
• 3 occurs twice, so P(X = 3) = 2/36 = 1/18
• 4 occurs three times, so P(X = 4) = 3/36 = 1/12
• 5 occurs four times, so P(X = 5) = 4/36 = 1/9
• 6 occurs five times, so P(X = 6) = 5/36
• 7 occurs six times, so P(X = 7) = 6/36 = 1/6
• 8 occurs five times, so P(X = 8) = 5/36
• 9 occurs four times, so P(X = 9) = 4/36 = 1/9
• 10 occurs three times, so P(X = 10) = 3/36 = 1/12
• 11 occurs twice, so P(X = 11) = 2/36 = 1/18
• 12 occurs just once, so P(X = 12) = 1/36

A Range of Values

We could also calculate the probability that a Random Variable takes on a range of values.

In other words: What is P(5 ≤ X ≤ 8)?

P(5 ≤ X ≤ 8) =P(X=5) + P(X=6) + P(X=7) + P(X=8)

= (4+5+6+5)/36

= 20/36

= 5/9

Solving

We can also solve a Random Variable equation.

Looking through the list above we find:

• P(X=4) = 1/12, and
• P(X=10) = 1/12

So there are two solutions: x = 4 or x = 10

Notice the different uses of X and x:

• X is the Random Variable "The sum of the scores on the two dice".
• x is a value that X can take.

Continuous

Random Variables can be either Discrete or Continuous:

• Discrete Data can only take certain values (such as 1,2,3,4,5)
• Continuous Data can take any value within a range (such as a person's height)

All our examples have been Discrete.

Learn more at Continuous Random Variables.

Mean, Variance, Standard Deviation

You can also learn how to find the Mean, Variance and Standard Deviation of Random Variables.

Summary

• A Random Variable is a set of possible values from a random experiment.
• The set of possible values is called the Sample Space.
• A Random Variable is given a capital letter, such as X or Z.
• Random Variables can be discrete or continuous.

Copyright © 2020 MathsIsFun.com