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This is the way I looked at it intuitively. Lets say you roll a pair of dice 10 times. There is a certain probability that the number you choose might be selected on each roll. I pick 2. Whoops. Only got 3 and 5. I pick (bet) different numbers on each roll. Now lets say I predict that a 2 might be selected at least once in the 10 rolls. Isn't the probablity of hitting a 2 once during 10 rolls higher than hitting it once an an individual roll? So I agree to play the lottery for the next 5 years off and on. Isn't the probablity of hitting my 6 numbers higher during this 5 year period if I play the same numbers all the time?

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I don't know if this is close enough to vote to close as duplicate, but Past coin tosses affect the latest one if you know about them? is very similar. –  MJD Feb 10 '13 at 21:16
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3 Answers

Nope, Look at the definition of Independent Events. Playing the lottery, like rolling dice, every event is independent of the previous, thus you could choose a different number, or keep the same one, it doesn't affect the outcome of the next draw

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Here's a smaller, simpler example of the same sort: We will flip a coin, three times. There are eight possible outcomes, each equally likely:

$$\begin{array}{cccc} & \text{First flip} & \text{Second flip} & \text{Third flip} \\ \hline\\ a & heads & heads & heads \\ b & heads & heads & tails \\ c & heads & tails & heads \\ d & heads & tails & tails \\ e & tails & heads & heads \\ f & tails & heads & tails \\ g & tails & tails & heads \\ h & tails & tails & tails \\ \end{array}$$

But looked at another way, there are three possible outcomes: three heads (a), three tails (h), or "mixed" two-and-one (b–g). And "mixed" is therefore three times as likely as the other outcomes.

Your idea is that you are more likely to win the lottery by adopting a mixed strategy for this reason, say by betting on heads, then tails, then heads again, rather than by betting on heads all three times. But this doesn't work. Both heads and tails are equally likely to come up on each of the three flips, just like every lottery number is equally likely to come up in each lottery drawing. If you bet on heads and heads in the first two flips, then the chance of the third flip being heads is still 50-50, regardless of the outcome of the first two flips; compare rows (a,b) of the table with (c,d), (e,f) or (g,h).

The short answer is that on the third flip, the coin doesn't care how you bet on previous flips, and it doesn't care what the outcome of the previous flips was. It doesn't care about anything, since it is just a coin. This is true even though the odds are strongly in favor of getting a mixed outcome overall, and strongly against all three flip coming up the same.

The lottery is the same: the little balls in the lottery cage don't care about your previous bets or the outcome of previous drawings, and this is true even though the odds strongly favor a mixed result, and are strongly against a result where the same number comes up every time.

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No, the probability of hitting any one set of numbers is no lower or greater than hitting any other set of numbers. The probability of hitting a set of numbers does not change if you play the same numbers every day or if you change them every day - because the probability of hitting any one set of numbers is equal to that of any other.

It is true that the probability of hitting your set of numbers does go up the more times you play (but it doesn't matter what set of numbers you play each time). Eventually you are almost guaranteed to win. But there is no telling how long it will take, how much money you have spent on tickets, or how big the pot will be.

So, yes, the probability of hitting your specific set of numbers is a lot higher if you play them every day for five years than if you played them only once.

What you are doing is commonly called the Gambler's Fallacy, http://en.wikipedia.org/wiki/Gamblers_fallacy

And for the sake of future reference, there is a difference between probability and odds, in a formal mathematical context. In fact, there is also a difference between them and likelihood. I recommend you stick with the words probability or chance.

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What are odds and likelihood? –  dfeuer Jan 9 at 11:21
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