Suppose Batman rolls a die 6 times. How many different outcomes are possible (Sample Space)?

Question

probability - statistics

Answer 1

"different' outcomes is ambiguous

it could mean every die roll has a different number.

IF so, then

2 rolls of a 2 sided die would have 2!=2 different outcomes, (1,2) and (2,1)

if there were 3 rolls of a die with just 3 different numbers total outcomes would be

1, if order didn't matter, but 3! = 6 if order mattered

312, 321, 213, 231, 123, 132

4 rolls of a 4 sided die would have 4! = 24 outcomes

5 rolls of a 5 sided die would have 5! = 120

6 rolls of a 6 sided die would have 6! = 720 different outcomes

roll a 6 sided die once, there're 6!/5! = 6 possible outcomes

roll a 6 sided die twice, there're 6!/4!=30

3 times 6!3!= 120

4 times 6!/2!= 360

5 times 6!/1!= 720

6 times 6!/0!= 720

7 or more times = zero as it's impossible to avoid at least one number repeating

but if you allow repeated numbers then

1 roll 6 outcomes

2 36

3 216

4 6^4

5 6^5

6 rolls 6^6= 46,656 different outcomes. that's if order matters

Answer 2

Two things to try when an apparently simple has you stumped. 1.) Make certain that you have correctly determined what the problem is asking. (This is called the 'translation' process. Taking the text of the problem and determining the mathematical relationships that are being asked for.) 2.) make a "toy" example of the problem, i.e., the simplest case you can think of, of the same phenomenon.

1.) You are being asked to find the size of the sample space for 6 rolls of a die. Each roll has 6 possible outcomes. Does order matter? Yes. If order matters, you are being asked for permutations.

2.) Simplest case. Let's pretend we had a 3-sided die and are going to roll it twice. Our outcome space is:

{1, 1}

{1, 2}

{1, 3}

{2, 1}

{2, 2}

{2, 3}

{3, 1}

{3, 2}

{3, 3}

We have 9, 3^2 possible outcomes. So, extend this reasoning to a six-sided die rolled 6 times.