daily4 lottery game

mrnerd

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I'm having trouble figuring out counting for probability questions. It seems that every problem I encounter is different and there's no algorithm to apply. As a result I can never really figure out how I'm supposed to apply formula's. This homework question I'm having trouble with is case in point, and perhaps someone could tell me where I keep going wrong with picking my counting technique.

In the Daily4 game, four numbers between 0 and 9 will be drawn in succession (repetitions allowed). The player marks four numbers on a game card and has a choice of how to play, straight (the player's numbers will match the four numbers drawn in exact order) or box (the player's numbers will match the four drawn in any order).
a. What is the size of the sample space? 10^4 = 10,000
b. what is the probability of a straight? 1/10,000 = 0.0001

I'm not understanding what I'm doing wrong here
c. What is the probability of a box if four distinct numbers are drawn?
4 distinct numbers = 10*9*8*7 = 5040
4 permutations of a "box" = 4P4 = 4
4/5040 = 1/1260
<-- I know this is wrongAnother way of doing the same problem:
each number is distinct: 10C4 = 210
1 combination of winning "box" = 1/210
<-- I'm also sure this is wrong

d. What is the probability of a box if two of the numbers drawn are the same?
pick 3 numbers, the remaining number has to be 1 of the 3 already picked.
10*9*8*3 = 2160
4 permuations of winning "box" = 4/2160 = 1/540

Thanks for your help!
 
I'm having trouble figuring out counting for probability questions. It seems that every problem I encounter is different and there's no algorithm to apply. As a result I can never really figure out how I'm supposed to apply formula's. This homework question I'm having trouble with is case in point, and perhaps someone could tell me where I keep going wrong with picking my counting technique.

In the Daily4 game, four numbers between 0 and 9 will be drawn in succession (repetitions allowed). The player marks four numbers on a game card and has a choice of how to play, straight (the player's numbers will match the four numbers drawn in exact order) or box (the player's numbers will match the four drawn in any order).
a. What is the size of the sample space? 10^4 = 10,000
b. what is the probability of a straight? 1/10,000 = 0.0001

I'm not understanding what I'm doing wrong here
c. What is the probability of a box if four distinct numbers are drawn?
4 distinct numbers = 10*9*8*7 = 5040
4 permutations of a "box" = 4P4 = 4
4/5040 = 1/1260
<-- I know this is wrongAnother way of doing the same problem:
each number is distinct: 10C4 = 210
1 combination of winning "box" = 1/210
<-- I'm also sure this is wrong

d. What is the probability of a box if two of the numbers drawn are the same?
pick 3 numbers, the remaining number has to be 1 of the 3 already picked.
10*9*8*3 = 2160
4 permutations of winning "box" = 4/2160 = 1/540

Thanks for your help!

One problem is that 4P4 is not 4! (Well, actually, it is 4!, but I was using "!" as punctuation ... what it isn't is 4.)

The question itself seems odd to me; the "ifs" could be taken as asking for conditional probabilities, or as changing the way the numbers are chosen from what had been stated initially. In the latter case, I'd expect you to have chosen your numbers accordingly, so it would change the problem.

But let's look at what you did in (c). Your denominator is the number of ways to choose 4 distinct numbers, counting order. (In fact, it's 10P4.) Your numerator is meant to be the number of choices that would count as a "box", since it would be the number of ways to arrange those four numbers. So that will be correct when you fix 4P4.

Your second try has as numerator the number of unordered choices for the 4 numbers, and the numerator is the number of those that you choose (1), so that, too, is correct. And when you fix the first, you will find that they agree.

I should say, this is correct IF you interpret the problem as supposing that you pick four distinct numbers because you know they will do so. I'm not at all sure of my interpretation.

Your counting in (d) is wrong; that's a tricky kind of problem,much harder than the rest. I'd rather not look at that one just yet.

First, can you confirm that you copied the exact wording of the problem, and correct it if you didn't? Also, I'd like to know whether this is part of an introduction to the idea of probability, as the first parts suggest, or if you are expected to know advanced ways of counting, such as the last part seems to require.

But I can agree with you that this sort of thing (combinatorics with even slightly tricky situations) requires careful thought, not routine applications of formulas.
 
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I actually don't know the answer to your questions. I'm going to the TA's office hours now to try and figure it out. I'll update when I understand.
I'm actually in a discrete math class, so this isn't a statistics class at all. We've just taken this weird little detour into counting/probability as part of our study on set theory. The counting techniques they seem to "expect" me to know seem quite advanced, but nothing was spelled out specifically.

thanks
 
I actually don't know the answer to your questions. I'm going to the TA's office hours now to try and figure it out. I'll update when I understand.
I'm actually in a discrete math class, so this isn't a statistics class at all. We've just taken this weird little detour into counting/probability as part of our study on set theory. The counting techniques they seem to "expect" me to know seem quite advanced, but nothing was spelled out specifically.

Actually, discrete math is just what I would expect you to be learning, if you are going at all deeply into combinatorics (counting), which is discrete math. (Probability often goes along with that, as an application; a statistics course commonly doesn't go into combinatorics.) It's not weird at all, just part of a broad collection of topics that seem separate, but are interrelated in many ways.

So my question will be, how "little" is this "detour"! Once we figure out what the problem means, I'll want to know how much of combinatorics you have learned, and whether some part of it is considered a prerequisite for the course.
 
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