Pulsified333
  • Pulsified333
involving the sum of the numbers showing on two fair dice. (1) What is the probability that exactly one die shows a 6 given that the sum of the numbers is 9? (2) What is the probability that the sum of the numbers is 9 given that exactly one die shows a 6? (3) What is the probability that the sum of the numbers is 9 given that at least one die shows a 6?
Mathematics
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SOLVED
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schrodinger
  • schrodinger
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Pulsified333
  • Pulsified333
@jim_thompson5910
jim_thompson5910
  • jim_thompson5910
which one are you stuck on? how far did you get?
Pulsified333
  • Pulsified333
i haven't gotten any yet. lets start with #1 @jim_thompson5910

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More answers

jim_thompson5910
  • jim_thompson5910
it tells us in #1, that `given that the sum of the numbers is 9` so we know, for a fact, that the two dice add to 9
jim_thompson5910
  • jim_thompson5910
what are the ways to add to 9 with 2 dice?
Pulsified333
  • Pulsified333
4&5
jim_thompson5910
  • jim_thompson5910
yep, what else? can you list ALL of the ways to add to 9?
Pulsified333
  • Pulsified333
3&6, 4&5
jim_thompson5910
  • jim_thompson5910
so put that all together we have 3+6 4+5 5+4 6+3 there are only 4 ways to do this
jim_thompson5910
  • jim_thompson5910
of these 4 ways, how many have exactly one "6" in them?
jim_thompson5910
  • jim_thompson5910
you sure?
Pulsified333
  • Pulsified333
2?
jim_thompson5910
  • jim_thompson5910
yeah 2 3+6 and 6+3
jim_thompson5910
  • jim_thompson5910
it would be a fraction Probability of exactly one 6, given sum of 9 = (# of sums with 6 in them)/(# of ways to add to 9) = 2/4 = 1/2
jim_thompson5910
  • jim_thompson5910
` given that exactly one die shows a 6` so we know die A is 6 or die B is 6 (both cannot be 6 at the same time)
jim_thompson5910
  • jim_thompson5910
if die A is 6, then what are the possibilities for die B?
Pulsified333
  • Pulsified333
5 and 4?
jim_thompson5910
  • jim_thompson5910
B could also be 1,2,3 basically it could be anything but 6
jim_thompson5910
  • jim_thompson5910
if die A is 6, then die B could be 1,2,3,4,5 similarly if die B is 6, then die A could be 1,2,3,4,5
Pulsified333
  • Pulsified333
but doesn't the sum have to be 9?
jim_thompson5910
  • jim_thompson5910
no, the given here is that one die is 6. That's it
jim_thompson5910
  • jim_thompson5910
this dice chart may help
1 Attachment
Pulsified333
  • Pulsified333
ok so die B could be 1,2,3,4,5
jim_thompson5910
  • jim_thompson5910
how many possible outcomes are there if exactly one die is 6?
Pulsified333
  • Pulsified333
but it says What is the probability that the sum of the numbers is 9
jim_thompson5910
  • jim_thompson5910
we'll get there
jim_thompson5910
  • jim_thompson5910
in the chart I posted, mark the row that has 6 and the column that has 6. But don't mark the cell that has both 6's. How many cells did you highlight?
Pulsified333
  • Pulsified333
5?
jim_thompson5910
  • jim_thompson5910
you should count 5 along the '6' row and 5 along the '6' column so 10 in total
jim_thompson5910
  • jim_thompson5910
how many of those 10, have a sum of 9?
Pulsified333
  • Pulsified333
2?
jim_thompson5910
  • jim_thompson5910
yes, 6+3 and 3+6 we have 2 sums that have a sum of 9 out of 10 outcomes so 2/10 = 1/5 is the answer to #2
Pulsified333
  • Pulsified333
oh ok, thanks
jim_thompson5910
  • jim_thompson5910
and for #3, it's almost identical to #2 BUT the key word in #3 is `at least` so it's possible to have both dice be 6
Pulsified333
  • Pulsified333
6&3 3&6
Pulsified333
  • Pulsified333
so is it 1/5? @jim_thompson5910
jim_thompson5910
  • jim_thompson5910
we have 2 ways to add to 9 (6+3 or 3+6) however, instead of 10 outcomes, we actually have 11. Again, it's possible to have both dice be 6 so it's actually 2/11
Pulsified333
  • Pulsified333
thank you :)
jim_thompson5910
  • jim_thompson5910
no problem

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