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Lottery Odds Question

1 points by tigertheory ↗ HN
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?

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?

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.

11 comments

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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.

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?
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?

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.

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