HOW TO: Probability Guide (When and how to use each type of probability)

[evandowsett]

Hi guys, I've decided to outline all types of probability and write up a guide on how to determine when you should use each one. I've spent a lot of time doing this for the many students who look like they need a hand with all this probability stuff! I've broken this post up into spoiler sections for ease of access. ENJOY 

The types of probability:

• Binomial Distribution
• Normal Distribution
• Continuous Density Functions
• Discrete Probability
• Conditional Probability
• Statistical Inference

Binomial Distribution

Spoiler

HOW TO IDENTIFY:
The first thing to realise is that Binomial Distribution is used with a series of Bernoulli Trials, where a Bernoulli Trial is something that only has two outcomes. For example, flipping a coin is considered a Bernoulli Trial as there are only two possible outcomes (heads or tails). Getting an answer right in a test is a Bernoulli Trial as there are only two possible outcomes (right answer or wrong answer) that are completely independent (one outcome has no effect on the next outcome). Be careful though, as there are some tricky ones to identify. Is "Rolling a 6 on a die" a Bernoulli Trial?? Yes. There only two outcomes (rolling a 6, or not rolling a 6). Let's look at an example.

Example:


Answer:
Let's figure out how we can determine what type of probability to use. Like I said, we have to first identify if there is a Bernoulli Trial here. Firstly, we know that the probability John hits the bullseye is 1/4. For each question you come across, ask yourself - "Are there more than two outcomes to this trial?" If there is only two, it will most likely be a Bernoulli. This trial only has two outcomes - hitting the bullseye or not hitting the bullseye. Therefore, we have a Bernoulli Trial and hence a Binomial Distribution question on our hands.

Binomial Distribution is largely done in your calculator. Searching for this example question, I looked at Exam 1s first and couldn't actually easily find one. They'll always be on CAS-enabled Exam 2.

So, there are a few aspects we need to consider in this question.
1. The probability that John hits the bullseye at least once from four throws
2. The probability that Rebecca hits the bullseye at least once from two throws
3. The ratio between these two numbers

Now that we've identified that it's a Binomial Distribution question, we can work this out. For Binomial Distribution, we always need p, n and x, where p=probability of success, n=number of trials, and x=how many successes you have. NOTE: x can be a range of values

John:

n = 4
p = 1/4
x = 1, 2, 3 and 4 (x>0) - We know this because we're looking for the probability that he hits at least one bullseye.

All this goes into our CAS (Binomial CDF). I believe this is MENU --> 5 --> 5 --> E

n = 4, p = 1/4, lower bound = 1, upper bound = 4

Therefore, Pr(x>0) = 0.6836


Rebecca:

n = 2
p = 1/2
x = 1 and 2 (x>0)

Into the CAS:
n = 2, p = 1/2, lower bound = 1, upper bound = 2

Therefore, Pr(x>0) = 0.75


Ratio:
A and D are clearly not correct. Rebecca has a higher chance than John, therefore the first number must be higher than the second. That counts out C. Then you just work out what the ratios are in a decimal form.

32/27 = 1.1852
192/175 = 1.0971
Rebecca/John = 0.75/0.6836 = 1.0971

Therefore, the answer is E (192:175)

These questions can be quite simple once you get to know them. All you need to remember is this: If the question refers to a trial that only has two outcomes, it is a Binomial Distribution question.

Normal Distribution

Spoiler

HOW TO IDENTIFY:
The question will say "Let the random variable X be normally distributed with a mean of... etc". It is very easy to identify a Normal Distribution question, because the question will state that it is one. NOTE: Normal Distribution follows the 'bell-curve'. Let's look at an example.

Example:


Answer:





Identifying Normal Distribution questions is quite easy, because the question identifies it for you. Try to develop a solid understanding of the bell curve and how the standard deviations/area work together, and this will make these questions much easier.

Continuous Density Functions

Spoiler

HOW TO IDENTIFY:
Like Normal Distribution, Continuous Density Functions are easy to identify. Many questions will actually state "A continuous random variable, X, has a probability density function given by..." so it is simple to tell when to use this type of probability. Whenever you see a function set out as in the example below, use continuous density function probability.

The biggest thing to note with this type of probability is that the area under the graph equals the probability of an outcome lying between two x-values. Integration is a large part of this type of probability. The total area under the graph is equal to 1 and f(x) is always greater than or equal to zero.

Example:




Answer:




Like I said above, make sure you understand that the area under the graph is the probability between those two points, so get comfortable using integration in this context.

Discrete Probability

Spoiler

HOW TO IDENTIFY:
Again, many questions tell you that the variable X is discrete. You can usually tell if it's discrete if the question offers a probability table of some sort, or if it says "X is a discrete random variable..."

If there is a probability table, we need to use discrete probability.

Example:




Answer:

Keep in mind as well:
E(X) = sum of (x*probability) of each section of the probability table
Var(X) = E(X^2) - [E(X)]^2
SD(X) = square root of Var(X)
The sum of all probabilities in the table must add up to 1.

Conditional Probability

Spoiler

HOW TO IDENTIFY:
Conditional probability is all about "If this happens, then this." If a previous outcome affects the next outcome, this is conditional probability.

Example:


We can see that this question refers to conditional probability, because whether she is fit or not in any given month has a direct effect on whether she is fit the next month. We always want to construct a tree diagram for these types of question as below. We know that, as per the question, in the first month Paula is unfit. This sets the scene for the rest of the tree diagram.

Answer:




When doing conditional probability, a tree diagram is the easiest way to actually understand what is happening. It is much too difficult to try and visualise situations like these mentally. So, if one outcome has a direct effect on the subsequent outcome, use a tree diagram to break up the information.

Statistical Inference

Spoiler

HOW TO IDENTIFY:
Whenever a question mentions either proportions, distribution of p^, margin of error, confidence intervals, a sample statistic or a population parameter, you are dealing with statistical inference (the new study design topic). This is pretty much your guide for recognising those types of questions.

Example:


Answer:


Remember that:
A confidence interval is the proportion of data that lies in the middle of the graph. This means that if you have a confidence interval of 95%, 95% of data is in the middle, which leaves 2.5% on each side. If you have a 99% confidence interval, this leaves 0.5% of data on each side.

How to work out z-score:


Margin of error: A margin of error is very difficult to define. It is explained in textbooks as “The distance between the endpoints of the confidence interval and the sample estimate”. It’s a bit ambiguous what this means, but you don’t really need to understand it. All you need to know is this:

That's really it for statistical inference!

Guys, have a read through the above sections and let me know what you think! I've put a fair bit of time into doing this so I hope it helps some of you out just post below if you have any questions about anything, more than happy to help!

[qwertyu1234567]

Guys, have a read through the above sections and let me know what you think! I've put a fair bit of time into doing this so I hope it helps some of you out just post below if you have any questions about anything, more than happy to help!

I honestly can't say thank you enough!! I really appreciate the time you took to do all of this

[evandowsett]

I honestly can't say thank you enough!! I really appreciate the time you took to do all of this

No worries qwerty, I hope you got some value from it!!

[Gromekk]

Just starting to read through it, looks great. Just thought you might want to add for Bernoulli trials the trials are independent (so that the outcome of any trial is not affected by the outcome of any previous trial).

Edit: Also for probability density functions, might be worth mentioning that the total area under the function is 1, and f(x) is greater than or equal to 0

[evandowsett]

Just starting to read through it, looks great. Just thought you might want to add for Bernoulli trials the trials are independent (so that the outcome of any trial is not affected by the outcome of any previous trial).

Edit: Also for probability density functions, might be worth mentioning that the total area under the function is 1, and f(x) is greater than or equal to 0

Thanks Gromekk, feedback noted and original post edited with that important information!

[Exhati]

THANK YOU SO MUCH FOR SHARING THIS!!!!

[halo]

THANK YOU!!!!

[brenden]

Smash that upvote button peeps! Good luck with your tutoring business Evan! (Link appears to be in his sig).

[evandowsett]

THANK YOU SO MUCH FOR SHARING THIS!!!!

THANK YOU!!!!

No worries guys! I'm glad it helped you out

Smash that upvote button peeps! Good luck with your tutoring business Evan! (Link appears to be in his sig).

Thank you for the support brenden! Appreciate it

[cookiedream]

Thanks for this guide, it's really comprehensive and super helpful
I think what's worth mentioning is how vcaa can sometimes be very tricky with conditional probability.

They often use words like "given ..." to present the condition, which is as obvious as "if".
But sometimes they also give a condition subtly in the form of a statement, before asking you for the probability.
Like in 2012 Exam 1 Question 4c (which we know only 3% of people got full marks in......)

[evandowsett]

Thanks for this guide, it's really comprehensive and super helpful
I think what's worth mentioning is how vcaa can sometimes be very tricky with conditional probability.

They often use words like "given ..." to present the condition, which is as obvious as "if".
But sometimes they also give a condition subtly in the form of a statement, before asking you for the probability.
Like in 2012 Exam 1 Question 4c (which we know only 3% of people got full marks in......)

Hey cookiedream, thanks for the feedback, I'm glad you liked it!

That's definitely true, thanks for that gold nugget of advice. It's sometimes difficult to actually determine what the condition is so just be careful about what it is the question is asking

[evandowsett]

Just thought I'd bring this to the top for you Methods students who are currently going through probability

Enjoy.

[kiki.]

Thank you so, so much!

[evandowsett]

Thank you so, so much!

You're welcome kiki! Glad to be of help. Good luck for your exams