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Probably the Tea Park one. I'm just talking through some slides I used in one of the lessons
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for the sake of those who weren't there. At various points you're probably going to want
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to pause this screencast and do some of the questions out of the note. We'll be looking
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at the probability scale, expected frequencies. Those questions about bags of balls being
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shaken up and things picked out of them. That brings us on to the idea of mutual exclusive
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events, the use of possibility space diagrams and then the important idea of independent
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events. Part two we'll look at tree diagrams.
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Problematic scale. If you stop people on the streets and ask them questions about coins
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and dice, they'll probably get the answer right. So the chance of getting ahead on a coin
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is 50-51-2. But in maths we prefer to talk about things in a more precise way. So we talk
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about the probability of an event being a fraction on a scale from north where it won't
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happen to one which is complete certainty. Most things are in the middle somewhere. When
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I say a fraction it can be a percentage, it can be a decimal or it can be a fraction
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with a top and a bottom like a proper fraction. The formula that can be used to calculate
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probabilities in simple cases with things like coins and dice and cards is simply to take
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the number of ways you win or the number of design outcomes and divide it by the number
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of equally likely possible outcomes. So for instance in the case of the coin there's
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one way of getting ahead so the top of the fraction is one and there are two possible
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outcomes so the bottom of the fraction is two.
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Talking of coins, Percy Diaconis, a professor of mathematics in New York has discovered
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that if you toss a coin heads up, catch it, there is a very small possibility that it'll
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lend heads up, very small extra probability that'll lend heads up. It's a very small effect
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but he's done the analysis and it's actually quite fascinating.
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Expected frequencies. The expected frequency of something happening is the probability
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of that event times the number of trials so if you toss a coin a hundred times you'd
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expect to get 50 tails on the coin. It doesn't always work if I simulated on a spreadsheet
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here as you can see when we actually do the experiment most of the time it's close
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to 50 heads but not quite. If I do a whole run of experiments I actually use 30 coins
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on this experiment. You can see that most of the outcomes are fairly close to 15 heads
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which is what you'd have expected but there are some quite large deviations or divergences.
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So observed frequencies aren't always the same as expected frequencies.
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And now onto the bags of balls questions which examiners love so much. Maybe because it's
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an easy example of a random process. You've got five red balls, two blue balls and three
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green balls in a bag. You shake up the bag, you can't see in the bag, you shake the bag
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up. Somebody else picks one at random. It's the lottery idea. What's the probability of
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getting a red one? Well there are five ways of getting a red one going back to that fundamental
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formula and there are ten possible outcomes you add up all the balls in the bag. So it's
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five tenths or a half. What's the probability of getting not a green one? Well that's quite
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clever. You can start saying, well I've got two red ones, sorry five red ones, two blue
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ones that makes seven not green. So that's seven tenths. Another way of working it out
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which could be useful is to say well something's got to happen so all the probabilities add
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up to one and I've got a three tenths chance of getting a green one. So I must therefore
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have a one minus three tenths or seven tenths chance of getting a not green one. That can
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be useful in certain circumstances. What's the probability of getting a yellow one? Well
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zero because there isn't a yellow one in the bag. You might get little trick questions
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like that on some of the practice material we use but you'll never get an exam question
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based on anything like that. Suppose you picked a green ball and then did not put the
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ball back in the bag you kept it on the bench. That alter some of the probabilities so now
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there's only nine balls left. So what's the probability of getting a green one again? Well
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you've got nine balls but there's only two green ones left as you took one of them out.
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So that's two ninths. What's the probability of getting a red ball? Well the five red
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balls are still there because the first one you picked out was a green so that's five
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over nine. You can get quite subtle little questions like that. So I think you better
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do some practice. Exercise one unit 19 page three. Exercise two unit 19 page five and
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remember that the probability of something not happening is one minus the probability
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that it happens. That can come in quite handy. You might want to pause the screencast now.
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Wait for the next until you've done the exercise before looking at the next bit.
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The next exclusive event sign normally explain music exclusive events by using an old fashioned
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telly. Remember the days when you had to get up to change the channel. There are four
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push buttons and one would be pressed in say BBC one. When you wanted to change to BBC
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two you had to press that button and the first one popped out. That gives the idea that
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you can only have one alternative of the four or five or six each exclusive event. Another
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example would be there's three sandwiches left in the canteen. Egg, cheese, chicken. You
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have to pick one of them but you can only pick one of them so he has to be either egg or
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cheese or chicken. When you hear yourself saying the word or that's often an indicator of
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mutual exclusive events. The probability of A or B you can find by adding up the probability
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of A and adding up the probability of B. As a concrete or means adding probability always
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as a concrete example. Suppose you wanted to know the probability of rolling a four or
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a six on a die. You would take the probability of the four which is a six. Add it to the
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probability of the six which is also a six. You get two six which cancels down to a third.
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That leads on to the idea of possibilities based diagrams. Suppose this is best demonstrated
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by a concrete example. Suppose you had two dice, one red, one blue. You add the scores
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on each dice to find your total score. There are 36 equally likely outcomes. Suppose you
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got one on the red you can have one, two, three, four, five or six on the blue. Same if you
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get two on the red. So for each red score there are six possible blue scores. Six six
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is a 36 but there's only 11 different scores because the smaller score you can get is one
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plus one and a larger score is six plus six or twelve. So there are going to be different
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numbers of different scores aren't there? The situation is best displayed in a table.
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See how I've made a table with the red dice along the top? Blue dice down the side and
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the black numbers inside the table are the different possible scores. Straight away there
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are six ways of getting seven. So that means that the probability of getting a seven score
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when you're old two dice is six over 36 which cancels down to give a six. Now on an
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exam paper you might have a partially completed table like that to fill in. Then they can start
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asking you questions like what's the probability of getting a score bigger than eight? Well four
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plus three plus two plus one I make that ten. So that's ten over 36 which is five eight
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eights. They can also ask you questions that loop back to your knowledge of maths so they
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might ask you what's the probability of getting a prime number score? So you have to remember
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what a prime number is. Well I record there are fifteen prime numbers between two and in
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that table because seven counts six times of course not fifteen different prime numbers.
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So the probability is fifteen over 36 which cancels down to give five to else. Looking
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now independent events. An independent event is when the outcome of one trial cannot possibly
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affect the outcome of another trial. For example tossing a coin or rolling a dice or indeed
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rolling two dice like the situation we've just looked at. When you tossing a coin or rolling
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a dice the probability of getting a head is a half and the probability of getting say a
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five is one sixth. When you multiply them together you get one twelfth and that expresses
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the idea that it's harder to get both a head and a five that's one outcome at a twelfth
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possible outcomes as opposed to just sort of getting a head. So and means multiply in
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probability. It has been suggested that there aren't any actual independent events in
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nature. Edward Lorenz whose photographs there asked the question does the flap of a butterfly's
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wings in Brazil set off a tornado in Texas and surprisingly enough the answer is probably
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yes. Just do a web search on butterfly effect if you're interested that's not on the GCSE
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exam. In summary the expected frequency is the probability of the event times the number
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of trials you do. It may be different to the observed frequency. All means add and you
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got mutually exclusive events and means multiply when you've got independent events.
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And if you do some more work on page five unit thirty seven you'll find some relatively
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challenging questions which test in all of this. And that brings us to the end of part
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one of probability.