Lottery Odds Question

#1

Post by tigertheory » Wed, Jun 10, 2020, 4:54 AM UTC

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.

Re: Lottery Odds Question

#2

Post by ColinWright » Wed, Jun 10, 2020, 5:12 AM UTC

> 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.

Re: Lottery Odds Question

#3

Post by tigertheory » Wed, Jun 10, 2020, 5:23 AM UTC

ColinWright wrote: post #2

> 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.

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?

Re: Lottery Odds Question

#4

Post by ColinWright » Wed, Jun 10, 2020, 5:32 AM UTC

tigertheory wrote: post #3

ColinWright wrote: post #2

> 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.

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?

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.

Re: Lottery Odds Question

#5

Post by tigertheory » Wed, Jun 10, 2020, 5:40 AM UTC

ColinWright wrote: post #4

tigertheory wrote: post #3

Earlier quoted context omitted.

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?

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 probabi…

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?

Re: Lottery Odds Question

#6

Post by ColinWright » Wed, Jun 10, 2020, 5:41 AM UTC

tigertheory wrote: post #5

ColinWright wrote: post #4

Earlier quoted context omitted.

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 probabi…

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?

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".

Re: Lottery Odds Question

#7

Post by tigertheory » Wed, Jun 10, 2020, 5:52 AM UTC

ColinWright wrote: post #6

tigertheory wrote: post #5

Earlier quoted context omitted.

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?

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".

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?

Re: Lottery Odds Question

#8

Post by ColinWright » Wed, Jun 10, 2020, 5:56 AM UTC

tigertheory wrote: post #7

ColinWright wrote: post #6

Earlier quoted context omitted.

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".

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?

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.

Re: Lottery Odds Question

#9

Post by [deleted] » Wed, Jun 10, 2020, 6:01 AM UTC

ColinWright wrote: post #6

tigertheory wrote: post #5

Earlier quoted context omitted.

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?

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".

[deleted]

Re: Lottery Odds Question

#10

Post by tigertheory » Wed, Jun 10, 2020, 6:07 AM UTC

ColinWright wrote: post #8

tigertheory wrote: post #7

Earlier quoted context omitted.

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?

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.

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.