
Lottery Odds Question - tigertheory
In a lottery that let’s you choose numbers between 1-70 for the five “white” balls, and numbers between 1-25 for the sixth “mega” ball, if after being drawn the white balls are ordered from least to greatest, what is the probability of getting exactly the lowest two white balls?<p>For example, let’s say the winning numbers for the first five white balls when ordered from least to greatest is 1,2,3,4,5 and the sixth mega ball is 10. What is the probability of getting exactly the lowest two white balls 1 and 2 correct and no other numbers?<p>Note that this is different than the probability of getting any two numbers to match- this is the probability of getting the lowest two numbers to match.
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ColinWright
> _what is the probability of getting exactly the lowest two white balls?_

It depends on how you choose your numbers. We can assume the numbers drawn in
the lottery are random, but the odds of you correctly getting exactly the two
smaller numbers and no others depends on how you have selected the numbers
against which you are matching those drawn.

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tigertheory
Let’s say that the first five white balls are pulled from the first container
containing 70 balls and that the sixth ball is pulled from a second container
containing 25 balls. What is the probability then of getting exactly the two
lowest numbered white balls from the first container?

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ColinWright
I still don't understand your question ... there are several possible
interpretations, and I don't know which one you think is the most obvious.

So here's one.

> There are white balls numbered 1 to 70.

> There are black balls numbered 1 to 25.

> I select 5 white balls uniformly at random without replacement.

> I select 1 black ball uniformly at random.

> I place all the balls in numerical order.

> What is the probability that the black ball is the third smallest?

But that description doesn't really seem to match the language you're using.
You talk about "Getting exactly the two smallest white balls right" ... I have
no idea what you mean by that. And what part is played by the black "mega"
ball?

If my framing is right (though I suspect it isn't) ... the first supplementary
question is: What if the selected white balls are numbered 1, 2, 3, 4, 5 and
the black ball is numbered 2?

By the way, the hardest part about these sorts of questions is learned how to
state them absolutely precisely. You will find that no matter how careful you
are, there will be someone who finds an alternate interpretation. I'm not even
convinced my statement above is completely water-tight.

Also, why are you asking? What's your application?

Finally, as I say, I suspect my framing is wrong, and that I really don't
understand what you're asking. If you're serious about this, you need to think
carefully about what's actually going on, break it down into very small steps,
and try to be absolutely precise about each step. So far, I suspect I'm not
going to understand your question well enough to be able to answer.

~~~
tigertheory
Let’s say W stands for white ball and B for black ball. Let’s say we draw and
order the white balls from least to greatest. It will look like this: W1 W2 W3
W4 W5 B1

I’m asking what is the probability I get the two lowest white balls W1 and W2
correct but get all other balls W3 W4 W5 B1 incorrect?

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ColinWright
What do you mean by "get W1 and W2 correct"? I get the bit about drawing the
balls and ordering them. I don't understand what you mean by "get them
correct".

~~~
tigertheory
That means that if those numbers are drawn randomly from the containers, I’m
asking what is the probability that a person guesses those numbers right
BEFORE they are drawn as happens in the lottery?

~~~
ColinWright
That depends on how the person chose their numbers. So now you need to specify
how the person chooses their numbers. Are they choosing their numbers
uniformly at random without replacement?

I still don't know that the black "mega" number has to do with this, and I'm
still curious as to where the question comes from.

~~~
tigertheory
Let’s say the person chooses those numbers randomly and in the same way they
are drawn (e.g. no replacement so number 1 can only be chosen once for the
white balls).

The black ball is there just to get the exact probability estimate for the
Mega Millions lottery.

To answer your question it comes from the Mega Millions lottery which draws 5
white balls and 1 black ball. And I’m curious to know what is probability I
get first two numbers right.

~~~
ColinWright
OK, so the question is this:

Suppose I draw 5 numbers uniformly at random from 1 to 70, and suppose I do so
twice. What is the probability that the smallest two numbers match, and no
others?

So let a1<a2<a3<a4<a5 be the number from one draw, and let b1<b2<b3<b4<b5 be
the number from the other draw. What is the probability that a1=b1, a2=b2, and
a3, a4, a5, b3, b4, b5 are all different.

Is that your question?

How accurately do you need to know the answer? You can get an approximation
quite quickly by simulation ... an exact answer will be horrible.

 _Edit: OK, I have answers, but it would be useful to know how accurately you
need your answer._

