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The Lotto 649 is a Canadian lottery where six different numbers are randomly
drawn from a set of balls with numbers 1 through 49. What are your chances of
holding a winning ticket?
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The number of different ticket combinations ( where you
can't use the same number twice) can be determined
by using:
_{49}C_{6 = 13,983,816}**

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So you have one chance in
13,983,816 of holding the winning ticket. The probability is 0.0000000715.
The more tickets you buy, the greater your change of winning... Suppose you buy
50 tickets (which will cost you $50 Canadian). Your chances of winning go up -
they're now a whopping 50 out of 13,983,816 - for a probability of 0.000003575
(but not much of an improvement).**

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What about the probability of winning anything?**

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1. The probability of matching NONE of the six numbers is:**

**
_{43}C_{6 }x _{6}C_{0 } over
_{49}C_{6
}
_{43}C_{6}
is the number of ways of choosing from numbers not on your ticket;**

_{6}C_{0} is the number of ways of choosing the six losers on your ticket;

_{49}C_{6} is the number of ways to choose 6 balls

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which works out to be 0.436, or 43.6%... which means that about half the time,
NONE of your numbers will match!
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2. Similarly, the probability of matching ONE of the six numbers is::**

**
_{43}C_{5 }x _{6}C_{1 } over
_{49}C_{6}**

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which works out to be 0.413, or 41.3%... the probability of matching ONE number!**

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3. The probability of matching TWO numbers is::**

**
_{43}C_{4 }x _{6}C_{2 } over
_{49}C_{6}**

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which works out to be 0.132, or 13.2%... the probability of matching TWO
numbers!**

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None of the above 3 results pays anything at all (i.e. matching none, 1 or 2)...
If you add the 3 probabilities, you get 98.1%...
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Which means that 98% of the time, you won't win anything!
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Enough already! What about the winning side of things?**

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The first winning ticket is the one where you match THREE numbers. The
probability of this occurring (using the method above once again) turns out to
be 0.018 or 1.8%... or about 2 out of 100... or about 1 out of 50. This means
that you must spend about $50 (on average) to hope to win $10.
Getting the picture yet? Your chances of winning the big
prize are next to nothing, and the chances of winning even $10 are only
one in fifty... Yet people win all the time because millions of tickets are sold
every week - and somebody has to win the prizes. It's just not very likely that
it will be any one person in particular - and that includes you!
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