Question

There is a die with two of its sides painted green. It is rolled three times. What is the probability that you will get a green at least one of the three times?

Answer 1

We have two sides painted green. That is we have two possible outcomes , we will get green or we will not get green. If we have only two possible outcomes then we use Binomial probability distribution to find out probability.

Formula for Binomial probability distribution:

P(X) =    

Where,

n = fixed number of trials,

x= total number of success out of n trials.

p = Probability of success

q = probability of failure.

p + q = 1, p and q are always between 0 to 1.

In binomial trial only two outcomes are possible. Success and failure.

Probability of success + Probability of failure = 1.

p = probability of success = probability of green side = 2/6 = 1/3

p = 0.33

And q = Probability of failure = probability of not getting green

                = 1- p

= 1-0.33

q = 0.67

Total number of trials = n = 3, p = 0.33 , q= 0.67

We have to find probability of at least one time green , that is ,

P(X = 1) + P(x= 2) + P(x=3)

X = number of times green occurs

We can write it as

probability of at least one time green = 1- P(X = 0 )

Put these values in binomial distribution formula:

P(At least one time green ) = 1- P(X = 0 )

=1-

                                                                         Formula for combination =

= 1- *  

=1- 1 *  

=1- 0.300763

P( At least one green) = 0.699237

P( At least one green) = 0.6992

The probability that we will get at least one green side is 0.6992

You can also find this probability P( X <1) using excel command “=BINOMDIST(0,3,0.33,TRUE)”

Please note that excel gives probability values for less than cumulative values.