Year 10 Maths WHAT ARE THE CHANCES?

WHAT WOULD YOU DO WITH A MILLION DOLLARS?

Invest in the fishing industry!

HOW MIGHT YOU FIND YOURSELF WITH THIS AMOUNT OF MONEY?

These people won 2.4 million pounds…….. dg&list=PL42E007F2014C7B7D

This man won $168 million list=PL435B69FA5078E4D3

Question Do these people live ‘happily ever after’ as millionaires? See what this man has to say… UI&index=13&list=PL42E007F2014C7B7D

WHAT ARE THE CHANCES? Understanding the odds of winning the lottery: list=PL435B69FA5078E4D3 list=PL435B69FA5078E4D3

Questions Knowing the odds can be useful in winning games such as lottto – are there other games that involve chance? One form of lotto requires choosing 6 numbers out of 49 – do you think there is a good chance of winning? The chances of winning are: 1 in 13,983,816

REALITY CHECK Statistics professor Stephen Clarke, from Swinburne University, calculated the odds of hitting the big one back in "The average Australian, even if they buy their ticket a couple of hours before the draw is made, they've got more chance of dying before the draw is made than they have of actually winning the first prize," Dr Clarke says.

WHO WANTS TO MAKE HISTORY? 80,658,175,170,943,878,571,660,636,856,403,766,975,28 9,505, 440,883,277,824,000,000,000,000. To give you an idea of how many that is, here is how long it would take to go through every possible permutation of cards. If every star in our galaxy had a trillion planets, each with a trillion people living on them, and each of these people has a trillion packs of cards and somehow they manage to make unique shuffles 1,000 times per second, and they'd been doing that since the Big Bang, they'd only just now be starting to repeat shuffles.

STATEMENT OF INQUIRY Logic and modeling can change our justifications for decision-making.

Question: How can knowing about probabilities help us in different areas of our lives?

THE STOCK MARKET

GUIDING QUESTIONS How can people use information about the likelihood of events to guide their decisions? How effective is it to use probability and modelling to help us to make judgments about future events?

THE BASICS: How is the probability of success or failure for a particular situation calculated? The following activity will help us discover this:

Activity Materials per group: One pack of 52 cards (remove jokers) Two A3 pieces of paper Marker pens

Instructions 1)Sort the cards into suits: Hearts Diamonds Clubs Spades How many cards are there in each suit?

Instructions Sort each suit in the following order: Ace, two, three, four, five, six, seven, eight, nine, ten, jack, queen, king Questions: 1)How many cards are there in each suit? 1)How many ‘face cards’ (jack, queen, king) are there in each suit? 1)How many twos are there in a pack of cards?

Probability with cards Instructions: Problem 1: Put all the spades to the left side of your piece of paper. Put all the red cards to the right of your piece of paper. Use this to do a Venn Diagram on the paper showing the events of S = drawing a spade and S = drawing a spade and R = drawing a red card R = drawing a red card (include the number of cards for each event inside each circle)

Venn diagram E = 52 RS In your groups work out and write down on your paper: Pr(a spade) = Pr (a red card) = Pr (a spade or a red card) = Pr (a spade and a red card) = 13 (Clubs)

Another question Instructions: Problem 2: Put all the spades to the left side of your second piece of paper. Put all the face cards to the right of your piece of paper. Is there a problem? Now do a Venn Diagram on the second paper showing the events of S = drawing a spade and F = drawing a face card

E = 52 SF 1093 In your groups work out and write down on your paper: Pr(a spade) = Pr (a face card) = Pr (a spade and a face card) = Pr (a spade or a face card) = 30

Probability Review Ex 1.1 of textbook

What are Mutually Exclusive Events? If two events cannot occur at the same time they are called mutually exclusive events. Addition Law for Mutually Exclusive Events: If A and B are mutually exclusive events then: Pr(A or B) = Pr(A) + Pr(B)

Mutually Exclusive Events How do we know that events A and B are mutually exclusive? Because there is no overlap.

Mutually Exclusive Events P(A ∪ B) = P(A) + P(B) P(A or B) = P(A) + P(B) or

Are these events mutually exclusive?

Another situation Here is a Venn diagram showing the numbers of students who do archery event (A) and badminton (event B): E = 100 AB Question: if we know that a particular student does badminton, how does this affect the probability that they also do archery?

E = 100 AB How many play badminton? 30 How many of these also do archery? 16 So what we need to work out is the probability that a student does archery, given that they do badminton. Pr( student does archery given they play badminton) = n(archery ∩ badminton) n(badminton) = This is called conditional probability because there is a condition on the outcome of event A – ie it depends on the condition imposed by outcome B

Conditional Probability In probability terminology: Pr(A\B) = Pr(A B) Pr(A\B) = Pr(A ∩ B) Pr(B) Pr(B) 16 = = = 8 15

Conditional probability with tree diagrams A company has two factories where it manufactures sports shoes. It is known that 65% of the shoes are made at factory A. It is also known that 10% of the shoes made at factory A are faulty, but the fault rate at factory B is only 7%. a) Draw a tree diagram for this information A B F F’ F b) If a pair of shoes is randomly selected from the company’s stocks, find the Pr (the shoes are from factory A, given that they are faulty) First we need to find the sample space – all faulty shoes: Pr (AF) = 0.65 x 0.1 = Pr (BF) = 0.35 x 0.07 = Now find Pr(A ∩ F) = Pr(A\F) = Pr(A ∩ F) Pr(F) So Pr(F) = = = = There is a 73% chance that the faulty shoes will be from factory A

How useful is this maths? Could this being useful to analyze real life situations? Who in particular might find this information useful?

Discussion: Is it reasonable to use probability information to predict human behaviour?