"Counting Rules" Problems

[RoboRob]

I was having trouble with two probability problems and was wondering if anyone knew how to do them.

1) A phone extension consists of 3 digits. If all the digits have a probablity of being selected, what is the probablity that the extension consists of the digits 1,2, and 3 in any order? Repetitions are allowed.



2)To win a lottery, a person must select 4 numbers in any order fro 20 numbers. Repetitions are not allowed. What is the probability that the person wins?

There are some multiple choice answers but I think it would be better without them.

[BruceZ]

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I was having trouble with two probability problems and was wondering if anyone knew how to do them.

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Yes, I do. Other people do too. /images/graemlins/wink.gif I'll just provide "help".


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1) A phone extension consists of 3 digits. If all the digits have a probability of being selected, what is the probability that the extension consists of the digits 1,2, and 3 in any order? Repetitions are allowed.

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I assume you mean that each digit has an equal probability of being selected. Note that each digit is independent, and compute the probability that each place is one of 3 digits.


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2)To win a lottery, a person must select 4 numbers in any order fro 20 numbers. Repetitions are not allowed. What is the probability that the person wins?

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Compute the total number of ways to choose the 4 numbers. One of these is the winner.

[uphigh_downlow]

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1) A phone extension consists of 3 digits. If all the digits have a probablity of being selected, what is the probablity that the extension consists of the digits 1,2, and 3 in any order? Repetitions are allowed.


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3 red balls and 7 blue. Chance of picking 3 red balls in the first three attempts with replacement.

I assumed you meant have an equal probability of being selected when you said "have a probability"

3*3*3/10*10*10

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2)To win a lottery, a person must select 4 numbers in any order fro 20 numbers. Repetitions are not allowed. What is the probability that the person wins?


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same as 4 red balls and 16 blu balls in a box. Pick all 4 red in 4 (not clarified) attempts.
you say repetetions. SO I guess its with replacement, in which every time a red is picked you replace it with a blue in the box

Answer should be 4*3*2*1/20*20*20*20

[BruceZ]

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2)To win a lottery, a person must select 4 numbers in any order fro 20 numbers. Repetitions are not allowed. What is the probability that the person wins?


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same as 4 red balls and 16 blu balls in a box. Pick all 4 red in 4 (not clarified) attempts.
you say repetetions. SO I guess its with replacement, in which every time a red is picked you replace it with a blue in the box

Answer should be 4*3*2*1/20*20*20*20

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No repetitions means no replacement, so that answer is incorrect.

[RoboRob]

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1) A phone extension consists of 3 digits. If all the digits have a probablity of being selected, what is the probablity that the extension consists of the digits 1,2, and 3 in any order? Repetitions are allowed.


[/ QUOTE ]
3 red balls and 7 blue. Chance of picking 3 red balls in the first three attempts with replacement.

I assumed you meant have an equal probability of being selected when you said "have a probability"

3*3*3/10*10*10

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2)To win a lottery, a person must select 4 numbers in any order fro 20 numbers. Repetitions are not allowed. What is the probability that the person wins?


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same as 4 red balls and 16 blu balls in a box. Pick all 4 red in 4 (not clarified) attempts.
you say repetetions. SO I guess its with replacement, in which every time a red is picked you replace it with a blue in the box

Answer should be 4*3*2*1/20*20*20*20

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1) The answer you get is incorrect. I was messing around and 3*2*1/10*10*10 gave me the right answer. I think this is because first you have 3 choices to get one of the 3 numbers.

After you get that first number, you need one of the remaining two numbers. Then you need the last number.

The answer is 0.006 in the solutions and thats what this gets you.

2)With no repitions I think the answer should be

4*3*2*1/(10*9*8*7) = 0.000206

The answer in the solutions is 0.0002 so I think this is correct because none of the answers in the choices go past that many decimal places, and they round.

[RoboRob]

Even though I got the solutions I appreciate the help. For some reason probability has been really interesting to me so I bought a copy of Probability DeMystified and have been doing the chapters in there. I'm past the first 6, only 6 more to go.

[uphigh_downlow]

no repetetions means repetetions are possible, but just not acceptable in your final solution which means replacements

Well I interpreted it that way anyway

[uphigh_downlow]

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1) A phone extension consists of 3 digits. If all the digits have a probablity of being selected, what is the probablity that the extension consists of the digits 1,2, and 3 in any order? Repetitions are allowed.


[/ QUOTE ]
3 red balls and 7 blue. Chance of picking 3 red balls in the first three attempts with replacement.

I assumed you meant have an equal probability of being selected when you said "have a probability"

3*3*3/10*10*10

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2)To win a lottery, a person must select 4 numbers in any order fro 20 numbers. Repetitions are not allowed. What is the probability that the person wins?


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same as 4 red balls and 16 blu balls in a box. Pick all 4 red in 4 (not clarified) attempts.
you say repetetions. SO I guess its with replacement, in which every time a red is picked you replace it with a blue in the box

Answer should be 4*3*2*1/20*20*20*20

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1) The answer you get is incorrect. I was messing around and 3*2*1/10*10*10 gave me the right answer. I think this is because first you have 3 choices to get one of the 3 numbers.

After you get that first number, you need one of the remaining two numbers. Then you need the last number.

The answer is 0.006 in the solutions and thats what this gets you.

2)With no repitions I think the answer should be

4*3*2*1/(10*9*8*7) = 0.000206

The answer in the solutions is 0.0002 so I think this is correct because none of the answers in the choices go past that many decimal places, and they round.

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Well its not a bad reverse engineering approach to try an find out what the question actually meant.


For question 1) you say repetetions are allowed. Now what does this mean? Can you have 111 as a acceptable extension. If so, then my answer is correct. If not... well you figured out the technique to correct it. Thats great. To be noted is that the qustion also says digist 1,2 and 3 NOT 1,2 or 3. So I do understand the answer could be 1*2*3/10*10*10

For question 2) You say repetetions are not allowed. Well it could mean two things. Repetetions are possible, but wont lead to winning the prize. or Repetetions wont ever happen.

Anyway I guess it was a test somewhere or a question somewhere online. A lil bit poorly worded, but I'm gald you have understoo the technique to solve them.

[BruceZ]

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For question 1) you say repetetions are allowed. Now what does this mean? Can you have 111 as a acceptable extension. If so, then my answer is correct. If not... well you figured out the technique to correct it. Thats great. To be noted is that the qustion also says digist 1,2 and 3 NOT 1,2 or 3. So I do understand the answer could be 1*2*3/10*10*10

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That is correct if valid extensions can have repititions, and we want one with all 3 numbers 1, 2, AND 3. I may have steered him wrong with my original help, since I assumed 1, 2, AND 3 with repititions allowed meant that the numbers just had to come from those 3 (which would really be OR).

[BruceZ]

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With no repitions I think the answer should be

4*3*2*1/(10*9*8*7) = 0.000206

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You mean 4*3*2*1/(20*19*18*17) = 0.000206 since there are 20 numbers. This is also 1/C(20,4).

[BruceZ]

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no repetetions means repetetions are possible, but just not acceptable in your final solution which means replacements

Well I interpreted it that way anyway

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It turns out that it means repetitions are not possible, which is the way lotteries with this many numbers usually work. You choose the numbers without replacement, usually by coloring them in on a form.

[RoboRob]

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With no repitions I think the answer should be

4*3*2*1/(10*9*8*7) = 0.000206

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You mean 4*3*2*1/(20*19*18*17) = 0.000206 since there are 20 numbers. This is also 1/C(20,4).

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Ya that's what I meant, sorry.

[RobbyD]

The lottery question is simple 20!/4!.

[BruceZ]

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The lottery question is simple 20!/4!.

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1/C(20,4) = 1/(20!/4!/16!) = (4!)*(16!)/20!

[RobbyD]

your completely right.. I lost the 16! in there.