Probability

Question

If you roll 5 six-sided dice what is the probability of rolling only one 6? what is the probability of rolling one or more 6's?

 

I don't understand how to find the total amount of outcomes much less how these two questions are different.

Answer 1

roll 5 dice

what is the probability of getting one 6 ?

how many ways can you get one 6 out of 5 dice ?

(5!)[1!(5-1)!]=5 ways

1 roll is a 6, and each of the other 4 rolls will be 1,2,3,4, or 5

5 ways *1*5*5*5*5=3125 ways to roll one 6 (out of how many outcomes ?)

the total number of outcomes is 6*6*6*6*6=7776 outcomes

P=3125/7776

 

what is the probability of rolling one or more 6s ?

this means what is the probability of rolling 1,2,3,4, or 5 6s ?

we already know there are 3125 outcomes for one 6

how many outcomes for two 6s ?

how many ways can you choose 2 out of 5 ?

(5!)/[2!(5-2)!]=5*4/2=10 ways

2 rolls will be 6s and 3 rolls will be 1,2,3,4, or 5

10 ways*1*1*5*5*5=1250 ways to roll 2 6s

so far we have 3125+1250=4375 favorable outcomes

how many outcomes for three 6s ?

(5!)/[3!(5-3)!]=10 ways

3 rolls will be 6s and 2 rolls will be 1,2,3,4, or 5

10 ways*1*1*1*5*5=250 ways to roll 3 6s

so far we have 4375+250=4625 favorable outcomes

how many outcomes for four 6s ?

(5!)/[4!(5-4)!]=5 ways

4 rolls will be 6s and 1 roll will be 1,2,3,4, or 5

5 ways*1*1*1*1*5=25 ways to roll 4 6s

so far we have 4625+25=4650 favorable outcomes

how many ways can we roll 5 6s ?

there is only one way that all 5 dice will be 6s

therefore, we have a total of 4650+1=4651 favorable outcomes out of a total of 7776 equally likely outcomes

P=4651/7776

another solution is to find the probability of rolling no 6s and subtracting this from 1

the probability of rolling no 6s is (5*5*5*5*5)/7776=3125/7776 (each of the five die will be 1,2,3,4,or 5)

1-(3125/7776)=4651/7776 which is what we got the first time

 

 

Answer 2

The total number of outcomes can be found by using the counting principle, which says that for each independent event in a series of events you can multiply the number of outcomes for each event to find the total number of outcomes.

In this case there are 5 independent events (each roll of the die), and since each one has 6 possible outcomes, we can multiply 6*6*6*6*6 to find out how many options there are (a lot!).  

 

The difference between the two questions is that just getting one 6 means that one time out of those five rolls you get a 6, and the other times you get something else (1-5).  Getting one or more 6's means that you have a few options: you could roll one 6, two 6's, three 6's, four 6's, or five 6's - but not zero 6's.  These are all separate probabilities that you will add up to find the total probability (or one that you can subtract to make things easier).

 

Anexample with 3 dice:  6 options for each roll, means that there are 6*6*6= 216 possible outcomes.

If i just want one of those to be a 6 then I have 3 ways to choose which roll will be a 6 (3C1 if you've done combinations), times 1 roll that has to be a 6 (1 option), and 2 rolls that can be a 1-5 (5 options).  If you guys have studied Binomial Probabilities, this is essentially using that formula.  Multiplying the options, we have 3*1*5*5 = 3*25=75 ways to roll a single 6 out of 216 possible outcomes.  So the probability is 75/216.

 

To get at least one 6, it's usually easier to find the probability of getting no 6's and then subtracting that from 1.  The probability of getting no sixes is the same as the probability of getting a 1-5 on every roll.  Which means that we have 5*5*5 ways of doing that, out of the 216 possible outcomes, which is 125/216.  Then we take that away from 1 to find out the probability of getting at least one 6, which would be 91/216.

 

Hope that helps.  It's a tough problem and I'm not sure what you're familiar with, so if you need anything explained in more detail or in a different way let us know!