You're positive

So you go to the doctor and get a test for a disease, and they tell you the test is 99% accurate. Then the phone call comes, your test came back positive. What is the probability you have the disease? Chances are it's not 99%.

If a test is said to be 99% accurate what they are referring to is the probability of the test coming back positive if you do have the disease. What you are interested in is the probability of having the disease if your test comes back positive; these are not the same thing. With all things that are not 100% accurate there is a chance of a false positive, being told you are positive when you are not, and a false negative being told you are negative when you're not.

Let's assume the disease is fairly rare and only occurs in only 2% of a population. For a population of the size of Canada, 24 million, that means 480,000 people have it and 23,520,000 don't. Since our test is 99% accurate that means that of the people that are positive 475,200 will come back positive and only 4,800 will be be told they are not, ie a false negative. At the same time, of the people that are not positive 23,284,400 will be told they don't have the disease and 235,200 will be told they do.

This leaves percetage of people that are positive out of the number that are tested positive is 67%. Since the disease is so rare the number of false positives is almost half as much as the number of people who actually have the disease, this makes telling a false positive from a real one very difficult. The rarer the disease the worse this effect. In many cases a test like this is used to screen patients since although it may scare a few people, very few sick people slip through the system. From there they can perform more expensive accurate tests to determine the results for sure.

If there's a 1 in 3 chance of winning, I'm guaranteed to win if I buy 3 tickets, right?

This year the Heart and Stroke Lottery had a 1 in 3 chance of winning. Actually they said 1 in 3, best odds ever. 1 in 3 is not actually an odds ratio, but the 1 to 2 odds are still the best odds ever. Odds are the ratio of wins to losses. Either way, technically there are 71,653 prizes and 250,000 tickets which are a 28.66% probability of winning, much less than the reported 33.33% chances of winning. But let us assume that the 1 in 3 is correct for the sake of simplicity. So you if bought 3 tickets at 1 in 3 odds you would have to win, right?

The probability of the first ticket wining is obviously 1 in 3. The probability of the second ticket winning is close to 1 in 3 but not exactly since the probability would have gone down in the first prize won, or up if it had not, but this is a tiny difference in 250,000 prizes so we'll ignore it. So by that the third ticket also has a 1 in 3 probability of winning. Since we are ignoring the effect of the missing tickets we are essentially assuming independence amoung the tickets, therefore we can multiply the three probabilities. That gives us that the probaility of just less than 4% of all three tickets winning. Which is not great, but it's not what we are looking for either.

So what about just winning one prize in 3 tickets. Well you do double your chances of winning, but those chances are still not that great. The probability of winning one or more prizes if you bought only three tickets = the probability of winning 1 or 2 or 3 prizes = 1- P(no prizes)=1-(2/3)^3. Which is about 70% chance of winning which is double the original 33.33%, but clearly not a guarantee.  If you double the number of tickets to 6, then you get all the way to 91% chance of winning a single prize. Of course the only way to guarantee you will in a prize is to buy 178348 tickets, one more than the number of non-winning tickets.

But then again the fact that you are supporting a good cause changes the game completely. This way you get to donate money to a good cause, gamble (which is always fun), and get a tax break all in one. What could be better. 

Gambling with Statistics vs Economics

I've heard a lot of colleagues say "As a statistician I cannot gamble", and they are completely right. As a statistician you should never gamble because statistically you will loose money on average. By that logic you can also never buy any type of insurance; in fact you could never really buy anything because on average you will loose money. What is important in all these situations is the utility or usefulness you get from the money you spent.

For instance if you buy a chocolate bar for a dollar you are loosing money in two ways, for the materials and for the labour. If you made it yourself it would cost a fraction of what you paid for it, but then again it would take time. Maybe to you an hour at your day job would make more money than the money you would save making a chocolate bar, so on balance you are saving money by paying for the time you saved.

When we gamble or buy something like insurance we are paying instead for pleasure or security (which in essence is a form of pleasure). Many things we pay for don't directly save us money like the chocolate bar but bring us pleasure and allow us to be more relaxed and therefore more productive, like going to a concert or playing a sport. Of course we can gain other types of utility from these activities lie networking and skills building, but they aren't really applicable to gambling. What it really comes down to is how much utility do you get from gambling? If you don't find it fun then don't go to the concert. 

The main problem of gambling is when the utility of playing surpasses the actual value to a person. If gambling is the one thing that makes you the happiest spending some of your income is no worse than getting a pass to the ski hills or playing paintball. Money in, pleasure out, that makes it possible to make more money. But just like any activity, if it completely drains your account and leaves you with nothing it's an addiction and does nothing to boost your morale. 

So what could be a better gift at christmas. Most children only spend a few minutes playing with most of their gifts anyway so why not buy them a lottery ticket. For many kids the thrill of a scratch ticket it equal if not more than a kinder egg, and they cost about the same, and that the lottery ticket has also a small probability of returning some or all of their money. Of course if you want a meaningful long lasting gift I would steer clear of the lottery tickets. It's all about the cost, is it worth it?

So instead of telling people how you don't gamble because it's statistically useless, tell them you take no pleasure out of gambling and therefore get no utility making it valueless to you. Because if some people didn't get pleasure out of gambling that first monkey would have never gambled leaving the trees in the first place