Expert: Jason Eisele - 8/21/2009

Question
Hi,

What would be the probability of rolling a 6 sided die say 8 times and getting a 6 once?

Can probability be more than one?

Thanks.

Answer
Hello Karrar,

Thank you for your question! The first step to answering it is to observe the probability of throwing a 6 on just one roll. Because there are six sides, and just one side has a 6 on it, you have a 1 in 6 (or 1/6) chance of rolling a six on any given roll. You have a 5/6 chance of rolling anything else on a given roll.

There are exactly eight different ways to roll a 6 exactly once out of eight rolls. If '6' means you rolled a 6, and 'X' means you rolled anything else, your sequence of rolls may look like any of these:

(6XXXXXXX)
(X6XXXXXX)
(XX6XXXXX)
(XXX6XXXX)
(XXXX6XXX)
(XXXXX6XX)
(XXXXXX6X)
(XXXXXXX6)

As you can see, there are eight possibilities, as defined by the mathematical function 'choose' by operating '8 choose 1' (you can type that right into Google! Try it with some other combinations!)

If you look at any one of these sequences, you will notice that each involves one occurrence of 6 (probability 1/6) and one occurrence of something else (probability 5/6). We can find the probability of any one of these sequences by multiplying the probability of each roll together. Each sequence has an occurrence probability of:

(1/6)*((5/6)^7) = 0.0465136079, or about 4.7%

Now, because there are eight sequences, each with this probability, we can simply multiply this by 8 choose 1. The full expression is this:

((1/6)((5/6)^7))*(8 choose 1) = 0.372108863, approximately 37.2%.

This percentage should seem reasonable, because if you had to guess how many times 6 would come up over eight rolls, one would be a pretty smart guess. This is because if you were to repeat this process several times, you would average 1 1/3 (or 8/6) 6's per every eight rolls. However, because of randomness of this small sample, you will divert from the average fairly often--a little more than 60% as you can see here.

See if you can play around and plug in some different numbers on Google Calculator to answer similar questions!

Now to address your simpler question, the probability of any given event cannot be more than one. A probability of 1 means than something is certain to happen--that there is zero probability of the event not happening. For this reason, there is no meaning of a probability greater than 1.