How would you find the odds of rolling a 4 with 3 dice?

Question

I know I should use permutations to do 6P3 to get 120 ways to roll 3 dice 

So I think I'm supposed to find out how many different ways you'll get a 4 with 3 dice but I'm not sure...

Answer 1

Chances to roll 6 on one die is "6".  if you have two dice the chances are "6 x 6".  If you have three die the chances are "6x6x6" or 216.  Similarly, the chances of you not rolling a six are "5x5x5" or 125.   Subtract 125 from 216 and you get 91.  So there is a 91/215 chance you will get a four using three dice.

Answer 2

There are 6X6X6 combinations of the faces of the 3 die.

To have a 4 on one, we have only

4XY

X4Y

XY4

where X,Y = 1,2,3,5,6

 

=> 3 X 5 X 5= 3 X 25 = 75

 

So P = 75/6X6X6 = 75/216

 

Odds in favor = Number of successes: Number of failures = # that have a 4:# that do not have a  4

= 75:216-75 = 75:141

 

Later it was said that there could be at least one 4.

 

Therefore

P = P(1 four) + P(2 fours) + P(3 fours)

For 2 fours:

44X

4X4

X44

=> 3 x 5 = 15

For 3 fours:

444

-> 1

75+15+1 = 91

So P = 75/216 + 15/216 + 1/216 = 91/216

Odds: 91:125

 

Answer 3

there are 6*6*6=216 sums when you roll 3 dice

there are 3 ways to get a sum of 4: 112, 211, and 121

the probability of rolling a 4 is 3/216=1/72

the odds of rolling a 4 are 3 to 213 because odds in favor of an event to occur is "the number of ways the event can occur"  compared to  "the number of ways the event cannot occur"

the event can occur 3 times and the event cannot occur 216-3=213 times