Since random variables can be used to form sets, it is possible to use these sets as an event space for a probability function:
Probability Using Random Variables
: The probability function which maps event spaces to probabilities
: The set of all elements with a random variable output of
: The probability of the event returned as number in the range
Many questions find probabilities using random variable sets. An example of this can be seen below:
Example of Random Variable Probability
Imagine a carnival game where we spin a wheel and are awarded tickets based on the area it lands on. We can create a sample space that contains each possible outcome as seen below:
Let us define the random variable which maps an area to the number of tickets you will be awarded
Note: The probability of the wheel landing on each event is proportional to its angle.
Probability Of Getting 10 Tickets
Probability Of Getting Less Than 5 Tickets
Alternate Notations
The probability that a discrete random variable is exactly equal to some value can be written in several different ways:
Probability Notation For Random Variables
A further explanation of each notation can be found below:
: This is the most verbose notation, however it clearly shows that a probability function takes in a set.
: This notation is very similar to the first notation, without the set brackets. We are still forming a set, but we choose not to write the set brackets to simplify the notation.
: This notation is usually referred to as the probability mass function as it takes a single numeric input and returns a single numeric output. The random variable is included as a subscript.
More information about the probability mass function can be found below:
Probability Mass Functions
A probability mass function ( PMF ) is a is a function which maps a random variable output to its respective probability. The input and output of a PMF are numeric, allowing us to treat it as a regular function.
Probability Mass Functions
It is important to realise that a PMF is still creating an event space and finding its probability, however this is all done behind the scenes. This means PMFβs can be used in a the same way as regular probability functions.
There are 3 rules which every PMF must satisfy. These are seen below:
Rules of Probability Mass Functions
Probability masses are always in the range 0 to 1
If is not a possible output of , its probability is 0
The sum of all possible probabilities is 1
Visualising Probability Distributions
A probability distribution allows us to quickly read the inputs and outputs of probability mass functions. This can be accomplished with tables or graphs.
Table Representations
If a discrete probability distribution only has a few outcomes, they can be easily summarised in a probability distribution table:
or
Note: The sum of probabilities in the bottom row must always equal to 1
An example of a probability distribution table can be seen below:
Probabilities as Tables
Suppose a random variable assigns a score to each marble in a game of marbles. The probabilities and scores for each marble are as shown in the diagram below:
This information can also be presented in a table:
2
3
4
5
6
12
14
18
21
34
0.21
0.28
0.05
0.06
0.11
0.17
0.08
0.08
0.03
0.03
Using this table, we can find a whole heap of interesting information.
Using the Table
Say we wanted to find the probability that we got a score below 5. We would start by highlighting all of the events in this event space:
2
3
4
5
6
12
14
18
21
34
0.21
0.28
0.05
0.06
0.11
0.17
0.08
0.08
0.03
0.03
Then to find the probability, we just add them:
So there is a 54% chance that we will score less than 5 points.
Graphical Representations
For a discrete random variable, we can plot the probability mass function as a bar or scatter plot. The x-axis gets the output of the random variable, and the y-axis gets the probability associated with that value:
These plots can be used to easily read off probabilities. An example of this can be found below:
Reading Distributions
You ask a person to pick a number from one to nine. The following distribution captures the probability that they will pick each number:
We can say quite a few things:
Individual Probabilities
The probability of selecting a 5 is given by:
So thereβs an 8% chance that a 5 is selected.
Multiple Probabilities
The probability of selecting an even number is given by:
So there is a 27% chance that an even number is selected.
Common Probability Distributions
Uniform Distributions
A uniform distribution has the same probability for every output with a non-zero probability. This follows the same rules that a uniform distribution would follow.
Uniform Probability Distributions
Where is the number of outputs with non-zero probabilities
If we know that a probability distribution is uniform, probabilities become quite simple to solve for:
Uniform Distribution Example
A coloured spinner with 16 equally sized parts is shown below:
A random variable is used to find the value of the spinner. Now since there are 16 equally likely outcomes, the probability of any particular outcome is just:
We could then find the probability that we hit a value higher than 12:
Now since all of them have a probability of , the probabilities are all the same, so we can directly evaluate the probability as:
Infinite Distributions
It is possible to have a non-zero probability for an infinite number of random variable outputs, this would lead to an infinite distribution:
But just remember, that the sum of all of these probabilities must be one:
A group of people jump along a line of platforms. 30% of jumpers fail to make it to the second platform. For each successive platform, 70% of the remaining jumpers successfully land upon it.
TODO: Show the jumpers in a line
The random variable tells us how many platforms a particular jumper managed to successfully land upon, so we get the following distribution:
We can use this distribution to solve a number of problems.
Particular Jumps
We can find the probability that the jumper will survive eight jumps by simply reading off the trend:
So only 2.47% of jumpers fall off after the 8th platform.
A Range of Jumps
We can find the probability that the jumper will survive between three and five jumps by adding the probabilities:
So approximately 32% of the jumpers will fall off between platforms 3 and 5 as weβve shown.
Outside a Range
We can also find the probability that a jumper will survive for more than three jumps by exploiting the complementary rule, which states:
But there are only three events in , so:
Therefore, there is only a 34% chance that a particular jumper will survive more than three jumps before falling off of a platform.
A video game maker has found that players select his characters according to the following probabilities . His characterβs stats are expressed by the following random variables:
Choose a random variable:
We can form an event space from the random variable by finding every element that obeys some condition:
Choose the type of condition
This event space can be used by a probability function to find the probability that a character in the set is picked.
Note: The set brackets inside of a probability function are optional Remove set brackets
Example Probabilities
There are a number of interesting probabilities that you can find by summing all of the probabilities in the event space. Try to find each of the following:
The probability of selecting a character with an
The probability of selecting a character with a
The probability of selecting a character with a
The probability of selecting a character with
The probability of selecting a character with
Feel free to try a few more of your own.
Difficulty
00
Time
SOLUTION
Difficulty
2 Mins
Time
Does the table below represent a valid probability distribution? Justify your answer.
1
5
8
12
0.75
0.05
0.14
0.06
SOLUTION
We can test if this is a valid probability distribution by checking if the probabilities sum to 1.
Hence this is a valid probability distribution
Difficulty
2 Mins
Time
1
5
8
10
0.7
0.1
0.14
0.06
A discrete probability distribution is given above. Find:
SOLUTION
Part 1
Part 2
There are no probabilities for , therefore the probability is:
Part 3
We can answer this question by summing the probability of all events above 3.
Part 4
We can sum the probability of all even events:
Difficulty
4 Mins
Time
Adam flips a coin three times. Let the random variable represent the number of heads he lands.
Using a tree diagram, list out the sample space
Use a probability distribution table to summarise the probability for each outcome of
Find
SOLUTION
Part 1
The tree diagram below represents all possible outcomes.
Therefore, we can write Adams sample space as:
Part 2
Since all events are equally likely we can write the probabilities as:
0
1
2
3
Part 3
We can use the probability distribution table above to get:
Difficulty
4 Mins
Time
Samantha is playing a game with her friends where she flips a coin and rolls a dice. Each outcome is then assigned a score based on the following rules:
Multiply dice roll by 1 if heads
Multiply dice roll by 2 if tails
Let the random variable represent the score of this game.
Use a tree diagram to list the sample space of this game
Assign each element in the sample space a score
Create a probability distribution table for the random variable
SOLUTION
Part 1
The sample space can be listed as:
Part 2
We can use the rules of the game to assign each outcome a score.
Outcome
H1
H2
H3
H4
H5
H6
T1
T2
T3
T4
T5
T6
Score
1
2
3
4
5
6
2
4
6
8
10
12
Part 3
The probability distribution table should contain all possible scores and their associated probability.
1
2
3
4
5
6
8
10
12
Difficulty
4 Mins
Time
1
2
3
4
5
2k
0.3
3k
3k
k
The table above is a probability distribution for the random variable . Find the value of k.
Hence find the value of .
SOLUTION
Finding k
We can find k by knowing that a net probability distribution sums to 1. We can therefore write the sum of probabilities as:
We can now solve for .
Finding The Probability
The probability of can be expressed as:
We can now substitute these values to get:
Difficulty
5 Mins
Time
1
2
3
4
5
The table above is a probability distribution for the random variable . Find the value of .
SOLUTION
We can find k by knowing that a net probability distribution sums to 1.
Simplifying gives:
We can now solve for using the quadratic formula:
We must ignore the value as it will result in events having a probability outside the range . Therefore, the value of is:
Note: If this value is substituted into any of the probabilities in the table they will all satisfy the condition
Difficulty
5 Mins
Time
Let each word in the sentence below be an element of the sample space .
βAll we have to decide is what to do with the time that is given usβ
If the random variable X represents the number of letters in an element, graph the probability distribution of .
SOLUTION
This sentence has 16 words, therefore we know that . We can then assign each outcome of a probability as seen in the probability table below:
2
3
4
5
6
We can then graph this probability distribution as seen below:
Difficulty
7 Mins
Time
Neymar, Messi and Ronaldo are each asked to kick a soccer ball at a target 40m away. The probability of each of them hitting the target is shown below:
Neymar: 72%
Messi: 84%
Ronaldo: 88%
Let the random variable represent the number of balls that hit the target. Use this information to create a probability distribution table for .
SOLUTION
Finding The Sample Space
We will begin this problem by drawing a probability tree diagram. This will help us list all the items in our sample space, as well as their associated probabilities:
Probability Of The Outcomes
We can then use the multiplication rule to get the probability of each individual event.
Outcome
Probability
0.532224
0.072576
0.101376
0.013824
0.206976
0.028224
0.039424
0.005376
Creating The Probability Distribution Table
The probability distribution table can be created by adding the probability of each scenario.
0
1
2
3
0.005376
0.081472
0.380928
0.532224
Difficulty
6 Mins
Time
The game CS-GO contains loot boxes which can be used to attain special items. The probability of attaining a rare item from these boxes is 5%.
Daniel has purchased 6 loot boxes. If he opens a loot box and it contains a rare item he will stop and sell the remaining boxes.
Let the random variable be defined such that:
Use a probability tree diagram to list all the outcomes of Daniels experiment. What is the associated probability of each outcome?
Use a bar chart to graph the probability distribution of the random variable
Find . What does this probability represent
SOLUTION
Part 1
W represents the event where Daniel wins the rare item and L represent the event where he didnβt. We can use this to draw the tree diagram below:
Note: Any time Daniel wins he stops playing, therefore that branch is terminated on the tree diagram.
We can list the probability of each outcome using the multiplication rule. This can be seen below:
Outcome
W
LW
LLW
LLLW
LLLLW
LLLLLW
LLLLLL
Probability
Part 2
We begin by summarising the probabilities of in a probability distribution table:
0
1
2
3
4
5
6
0.7351
0.05
0.0475
0.0451
0.0429
0.0407
0.0387
Note: Each value just comes from the probability of the outcomes in part 1
Graphing this on a bar chart shows:
Part 3
The easiest way to find is to find the complement of
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Probability Distributions
Cambridge Advanced Year 11 - 11 A
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4(a,d)
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8(c,d)
9(b,c,e)
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15(b)
Lesson Loading
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Forming Probability Distributions
Review Questions
What is ? Utilise the distribution shown in the video.
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Working With Probability Distributions
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Frequencies To Probabilities
Review Questions
How do we convert a frequency into probability?
Divide each outcome by 100
Divide each frequency by the total number of frequencies in the table
It depends, itβs different for every distribution.
Γ
Probability Mass Functions
Review Questions
What is a probability mass function?
A function which maps random variable outputs to probabilities.
A function which tells us the probability of our entire sample space.
A function which gives us the probability of events in our sample space.
Which of the following is an equivalent representation of
What does the x-axis of a probability distribution graph represent ?
The probability of the random variable
A random variable
The outcomes of a random variable
What is the sum of all outcomes in a probability mass function (pmf) ?
