# Probability favours worse odds?

1. Mar 11, 2015

### iDimension

This is a bit of a strange question but why does it seem like probability of something happening always seems to be less than the actual probability?

For example I am running a UK lottery simulator and you'll notice that the times I win is ALWAYS lower than what the probability says it should be, why is it never higher? Why does it never go higher but instead always lower?

In all 3 you see that the times I win is lower than the expected times to win, never does it go higher.

1 in 57 is higher than 1 in 59
1 in 1,033 is higher than 1 in 1,062
1 in 55,492 is higher than 59,984

Is it just a coincidence that the actual outcomes are always lower than what is expected? I've run the program 3 times now for a length of 15 minutes and the actual wins are always lower, never higher, why is this?

As the game plays on to infinity it will end up equalling out but in the early stages it just seems that the actual probability is always lower than the expected probability.

Last edited: Mar 11, 2015
2. Mar 11, 2015

### micromass

It should be a coincidence

3. Mar 11, 2015

### micromass

Oh by the way, if you just let your applet run then chances are very large that the probabilities will remain lower. This follows from the theory of random walks. If you actually want to see higher odds, then the best way is to restart the program a few times.

4. Mar 11, 2015

### iDimension

Yes I noticed when I restarted the program in the very very early stages it seems quite common to win more than the expected probability but once the game reaches 10,000 plays the probability changes to lower odds instead of higher odds. It seems strange but it would seem that the more you play the less likely you are to win the smaller prizes.

So you're best chance of winning a small prize is actually in the first 10,000 games, after that the odds of winning seem to be lower as the game goes on.

Last edited: Mar 11, 2015
5. Mar 11, 2015

### micromass

Those are the wrong conclusions as the odds of winning the game always remain the same. It's just that if the odds are lower than theoretical then they will remain lower for a long time. But conversely, if the odds are higher they will remain higher for a long time.

See http://www.math.uah.edu/stat/applets/RandomWalkExperiment.html (use n=100 and "Last zero"), you'll see that the total winnings (red line) doesn't change very often from winning to losing or from losing to winning. In fact, if you're balance is negative, then it will likely remain negative and the same with winning. This despite the fact that the chance of winning is always 1/2.

You would expect that the balance is positive half the time and negative half the time. This is in fact not so. Very counterintuitive.

6. Mar 11, 2015

### iDimension

But what I don't understand is why the odds are always lower? If it's true probability then you would expect to see slightly high and slightly lower odds but whenever I run it, the odds never seem to go lower than what is expected, just higher.

as you can see I let the program carry on running and after 330,000 games the odds are STILL lower, when if ever will they balance out or go lower? Why are they always higher? This is what I don't understand.

You might notice I did match a raffle and a 5 ball draw but for this particular match it's still the early game so like I said it's quite common to win early on with respect to the probability of that particular set.

I'd actually be surprised if I didn't win the jackpot before the 13,983,816th game. Just like I matched 3 numbers much more often than I 1 in 57 during the early game but as the game tends to infinity it seems the probability always favours slightly higher than the actual true probability.

I doubt that 1 in 58 will ever become 1 in 57, even after a trillion games.

7. Mar 11, 2015

### micromass

Restart the program 10 times or so and let it run for one or two minutes. Tell me if you still observe the same.

8. Mar 11, 2015

### Staff: Mentor

It could be a bug in the program or wrong odds given, but the given sample size is too small to check. Can you run it for much longer?

There could be a more subtle bug: the odds could be inclusive ("chance to get at least this"), while the win frequency seems to be exclusive.

The last picture shows an higher than expected rate of 5 correct numbers, by the way.

9. Mar 11, 2015

### iDimension

Unfortunately the program is quite slow so I couldn't run it for a very long time. I stopped after 1million games which took about 45 minutes. The odds are correct as I checked them on the official UK lotto website.

10. Mar 11, 2015

### DaveC426913

It would be interesting to take this up with the lottery corporation and see what they have to say. It could put them on-the-spot.

11. Mar 11, 2015

Staff Emeritus
Well, for the 50,000 GBP prize, your odds are higher.

For the 25 GBP prize, odds of 1:57 corresponds to a win frequency of 1/58.

12. Mar 12, 2015

### Staff: Mentor

What was the result? ;)

13. Mar 12, 2015

### mathman

If you take into account the one standard deviation range, you will find that the results are not so strange.

14. Mar 12, 2015

### DaveC426913

I don't see how it explains the consistency in bring low so often.

15. Mar 12, 2015

### iDimension

I forget what it was exactly but apart from the 6 ball match the others were still lower than the actual probability. The 3 ball match stayed at 1:58 and the 4 ball match ended up something like 1:1094 or something I can't remember exactly.

Yes standard deviation of one but why must that always be higher, for example why can't the standard deviation make it 1:56 instead of 1:58 or with the 4 ball match why is it 1:1094 instead of 1:1000? Just a bit strange that the actual probability is always lower, with the exception of the 6 ball draw of course.

The point I'm making is while you will sometimes get better odds and worse odds it seems to favour worse odds. Of course I only ran the program 3 times so I don't have a large sample to work with, it could just be that computer probability isn't 100% true probability perhaps?

16. Mar 13, 2015

### Staff: Mentor

That is not enough to draw conclusions. If we just consider "odds smaller or larger than predicted", we have 6 relevant data points (5 matches are too rare for conclusions), both with about 50% chance to be below and 50% chance to be above the expectation value. That gives a probability of about 1.5% to see all 6 data points below the expectation value.

17. Mar 13, 2015

### mathman

In general (for binomial distribution with small mean) the probability of getting less than the average is greater than the probability of getting more.

Example: Prob. of win =.01. 150 tries. Prob(0 wins) =.2215, Prob(1 win) = .3355, sum = .5570, even though expected no. wins is 1.5.

18. Mar 14, 2015