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Author Topic: A Guide on Probability in the HSC You Probably (Definitely) Should Read  (Read 14524 times)  Share 

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jamonwindeyer

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Hello again everyone, time again for one of the daily guides! This one will be on probability, a fairly small part of the course, so this is going to cover both 2 unit and extension probability. I'll hide the extension stuff away in a spoiler and save the 2 Unit Students some time... Use it to study guys  ;) I don't have to remind you that there are awesome notes available for both 2 unit and extension right here. Also, if anything in these guides if ever unclear (I try to get a balance between being succinct and being clear), shoot a question below! I'll be sure to post a response that will benefit everyone's knowledge. You can register for an account right here  ;)

Right, probability. This is a weird one. A lot of people I talk to think this is one of the hardest parts of the course. Others think it's easy. I would put it this way. The math itself is some of the easiest you'll come across in the exam. Understanding the question, however, is something different entirely. Most, nay, all of your study in this area, should concern interpreting the situation and applying the correct reasoning. Practice makes perfect here, let's look at a few scenarios, all adapted from past HSC questions.

Example 1 (2013 HSC): A bag contains 4 red marbles and 6 blue marbles. Three marbles are selected at random without replacement.
a) What is the probability that the first marble selected is red?

There should be no issues here, four out of the ten marbles are red.



b) What is the probability that all three selected marbles are red?
Slightly trickier, and the source of a common mistake. Remember that for every marble selected, we now have less marbles in the bag! Also remember the standard formula for the probability of events occurring simultaneously. We simply multiply the individual probabilities together.



c) What is the probability that at least one marble selected is red?
This one can either be very time consuming, or very easy, and it's a perfect chance to bring up complementary results. Think of them as opposites! The complementary result to drawing a red card in poker, is drawing a black card, for example. It's the event that doesn't occur, and it comes with a formula which makes this question a snap:



This stems from another useful fact; probabilities add to one! That is, the probabilities of every event happening, added together, always equal one. Remember this, it could come in handy, and is a useful way to check your answers.

Besides this, the only other thing to remember is the difference between mutually exclusive and mutually non exclusive events. Forget all the technical language... Mutually exclusive events are ones which don't overlap! For example, drawing a black card and drawing a diamond are mutually exclusive. Drawing a diamond and drawing a red are not, there is overlap! You have to take this into account in your questions.

Example 2 (HSC 1997): What is the probability that a single card drawn from a standard deck of 52 playing cards is either a red card or an even numbered card?

We have to consider the fact that some red cards are evenly numbered, and subtract this from our result. Don't count them twice!



If you are an extension student, we have a bit more to go:

Spoiler

The last piece of the puzzle for you guys is combinatorics. Better known as permutations and combinations. There is awesome notes on that here .

The main thing to remember is,  the number of permutation arrangements is order dependent. Combinations are not. I never did find any good tricks/ways to remember that, post any tips below! You also need to know factorials. These are explained in the notes, and are quite simple once you grasp the concept. The formulae are:



where n is the total number of elements, and k is the number of elements we want to arrange.

Applying these formulae is just substitution, so I won't give an example of that. I want to tackle this infamous question from the 2011 extension paper. For those who don't know, 2011's paper was infamously difficult and caused a bit of an uproar. This question was featured by the Sydney Morning Herald. Supposedly extremely difficult. Not for us!

A game is played by throwing darts at a target. A player can choose to throw two or three darts. Darcy plays two games. In Game 1, he chooses to throw two darts, and wins if he hits the target at least once. In Game 2, he chooses to throw three darts, and wins if he hits the target at least twice. The probability that Darcy hits the target on any throw is p, where 0 < p < 1.
(i) Show that the probability that Darcy wins Game 1 is .

This first part is simple use of the complimentary result theorem, don't be scared by the variable!



(ii) Show that the probaility that Darcy wins Game 2 is .
Similar method as above works a charm, but slightly trickier logic. We have to consider both when he doesn't hit at all, and when he hits once:



The last part of that expression might be a bit confusing. We consider the probability of one hit and two misses, that is:



However, that hit can occur on the first, second, or third throw. This tells us there are three possible arrangements if one hit occurs, all equally likely, so we multiply by three  and expand to get the total probability as appears above.

(iii) Prove that Darcy is more likely to win Game 1 than Game 2.
This is a bit of an algebra trick more than anything. Now normally this is a big no no, but start with your final result and work backwards:



which is true since 0<p<1, so the original condition holds. Note that this is a perfectly acceptable way to approach proofs in mathematics and extension, and indeed beyond. You just have to be EXTREMELY careful that you don't use any assumptions in your working, and are clear and logical in your argument.

See! Not such a hard question SMH  ;) Extension probability can be a doozy, but it is in my opinion, worlds better than projectile motion or binomial proofs. The last question of my HSC paper was probability, and I cheered. It's easy math! Just think carefully, and go slow, and the answers will reveal themselves. Oh! And remember that circular arrangements use (n-1)!, not just n!. Everyone else will forget it, you won't!


That's pretty much it for probability, as I mentioned, the math is really easy! But you have to be really careful in approaching probability questions, HSC feedback tells us they are messed up by a lot of people. Here are a few tips:

1- Biggest piece of advice you will ever receive... Tree diagrams are life! A good tree diagram virtually guarantees the marks for that question, they are super easy to interpret and ensure that you haven't missed anything. Check that every set of branches adds to one!
2- Some questions benefit from drawings also! Anything to get your head around the scenario.
3- To check if you've understood correctly, make sure the probabilities of all the scenarios involved add to one as required
4- A question which encourages use of complementary events is virtually guaranteed to be in your exam, it is almost always asked. Be ready!

That probably (see what I did there) wraps it up! Probability is a strange part of the course, and it's one of those bits for which practice yields dividends. Aim to have seen every possible question on marbles that there is before your HSC! Approach questions with caution, be clear with your logic, and TREE DIAGRAMS!

Be sure to pop any questions below if any of the working here seems confusing. You can register for an account here to continue sharing in all these useful guides and notes!
« Last Edit: October 03, 2017, 08:04:05 pm by jamonwindeyer »

arunasva

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Re: A Guide on Probability in the HSC You Probably (Definitely) Should Read
« Reply #1 on: September 08, 2017, 12:19:23 pm »
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T.This is so good, thanks
:3

bdobrin

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Re: A Guide on Probability in the HSC You Probably (Definitely) Should Read
« Reply #2 on: October 03, 2017, 08:00:21 pm »
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Hi Jamon,

Just wondering for the HSC 2u example 2, how did you find out that the probability of drawing an 'even-red' card is 5/52?

Thanks,
Ben

jamonwindeyer

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Re: A Guide on Probability in the HSC You Probably (Definitely) Should Read
« Reply #3 on: October 03, 2017, 08:03:05 pm »
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Hi Jamon,

Just wondering for the HSC 2u example 2, how did you find out that the probability of drawing an 'even-red' card is 5/52?

Thanks,
Ben

Hey Ben! Wow, surprised I missed this for so long - Should definitely be 10/52!! 2, 4, 6, 8 and 10 for diamonds and hearts - I'll tidy that up! :)

Thanks so much for the spot!!