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Medical Test: Conditional Probability, Bayes' Theorem


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assume that your doctor tells you that specific medical test came up positive – question – always – or are you not – it's not as simple – as it sounds – like – of – statistics – stochastics involved here – and so get some medical tests – for instance – this test could have – the following – properties – the probability – of the test given the positive result – when you is – could be for instance – ninety percent – on – in the probability – of the test – giving the negative results when you're real – is point number these two guys to some two number cost number hundred percent – either your tested positive or tested negative period – assume – furthermore – that – healthy people – are only tested positive – in – very very few – cases hopefully – going to see what this means – that say this mobility is – pronoun part in thousands – among – number thousand – and three persons – pronoun – baby tested positive – and the remaining – nine hundred and nineteen – nine – to be tested – negative – things – should be that this should be positive for – determiner person to be negative – for healthy persons – these quantities your conditional probabilities – is determiner condition given on the condition that you are – what's the vapidity – that you just – the data in the on the population – assume – that's – the probability – of somebody being healthy is – over point nine nine nine nine nine – so it's really determiner rare disease – in dealing with – the ability of somebody – being in – the opposite – number minus this number hundred percent minus ninety nine – point nine nine nine – percent – that's – point oh oh – oh – number – no – that's dude with the computation – with some actual numbers – of persons let's assume that you have determiner population – of – number hundred million – then – this fraction – is healthy so that would be ninety nine million – nine hundred ninety nine thousand – these are – the people – the rest – is in – these are not too many – it's just CO – point oh oh oh oh number – times number hundred million – number thousand – among – the ahead of the – people – how many are going to be tested – positive – and see – if you are and see – if you're healthy – they have determiner chance of – old point number percent – being tested positive so – opened number percent of the eessa tested positive which is ninety nine thousand nine hundred ninety nine – these – are false positives – they should not be tested positive – they should be tested negative because there is determiner syllabus force – positives – the rest – which is the majority luckily – is tested – negative – oh point nine nine nine nine nine times these ninety nine million nine ninety nine thousand – which is ninety nine million – eight hundred ninety nine thousand – and number – these are tested negative – this is as it should be – the test comes out negative – for – the people with some – exceptions – now the person s – how many are tested positive – ninety percent of the persons are tested positive – which is nine hundred if you start with number thousand – the – remaining number hundred – ninety number hundred which is these ten percent – off once negatives – the test comes out negative – but it shouldn't – these guys are really – now we can compute – the probability – that is of interest to us – namely – that's – another conditional probability – but it's the conditional probability – of – what's – the chances – are – you – being – ill – if you want tested positive – that way round – so before – the data that we had was the other way wrong what's the chances of being positive – if – you are determiner – no – condition – has changed – you – so – we can – do the numbers – it's on the condition – that you tested positive so – the only people that you have to look at uhhh these nine hundred – and these – ninety nine thousand nine hundred ninety nine – we have to check – which fraction – of these – is actually ill – and that's – these nine hundred – so – that's the probity – looking – for – how many – restriction pronoun should be saying which fraction of the persons were tested positive – are really – ill actually determiner – this fiction – and – since it dealing with the abilities they can do very course – computations – that say this is about number thousand – divided by – well – number hundred thousand – just the order of magnitude five – scans – three zeros – and then we get to this is pronoun over hundred – number percent open paren oh number – so – given – that you tested positive – your morbidity of being in his number percent – which – is not that much – of course that depends on the date of the tree – and that here – if these dates are different is the result can be pretty different to – know – the outcome but – something that's general interest in general – this formula looks like the following pronoun need to – decline – the – old – okay so what do we have in the – numerator – on these nine hundred twelve that they come from – this was the total number of persons – times – the probability – of being in – times the conditional probability – of being tested positive – if you are – so we've got – total number of persons – times – ability of – times – the conditional – probability – of being disappointed – if you are – same number PS – downstairs – copy – conditional probability – times the ability – times – the total number plus – and now these – ninety nine thousand nine hundred ninety nine – we – start with the total number multiply – the probability – of being healthy – and multiplied – by the condition of the ability of being tested positive – if you are – so this is the total number of times the probability – of being healthy – times the conditional – probability – of being tested positive – if you are – anything – and – we can cancel the – total number of persons you – can just get some phone line voting – some – sorts of probabilities – closely you see that the – abilities – occurring in this equation – after – class – on the left-hand side – they are being conditioned – on being ill – or being healthy – and our results is – just the other way around – period probabilities – the validity of being ill and it's conditioned on – being tested positive so this is an equation that allows us to invert the auto you – inside – the – condition of ability – this is called space – theater