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A fair six sided die is rolled four times in a row. What is the probability that die will come up six at least once?

2 Answers


#1/6+1/6+1/6+1/6=4/6=2/3# or #4(1/6)=4/6=2/3#


Each time we roll a fair six-sided die, there is a 1 in 6 chance that it will come up as a six. We can then use this to figure out what the chance is that a six will be rolled at least once over 4 throws. Because there is often confusion about whether to use addition or multiplication, let's do the math two different ways and see what happens:

The first way is to take each throw and see it as an individual event, each one having a 1 in 6 chance of getting a six, and that there are four of them:


The other way to view it is to say that there are four throws, each with a chance one in six chance of being a six:


How you think about the throws will determine how you do the math.

Sep 21, 2016


#1- 625/1296#



To get a 6 "at least once", means it can be

#rarr# Once in 4 rolls,or
#rarr# Twice in 4 rolls, or
#rarr# Three times in 4 rolls, or
#rarr# Four times in 4 rolls

There are obviously many combinations in each of these outcomes which will require lengthy calculations!

Note that the only outcome which is NOT included is

#rarr# No 6 in 4 rolls.

There is only ONE combination of this happening.

Not 6 and Not 6 and Not 6 and Not 6

#P(6) = 1/6 and P("not 6") = 5/6#

#P(N,N,N,N) = 5/6 xx5/6 xx 5/6 xx 5/6 = (5/6)^4#


The sum of all the probabilities is always equal to 1.

#("Prob of at least one 6") = 1 - P("no 6")#

#1- 625/1296#