I assume you have done binomial distribution before.
Binomial distribution applies when
- experiment consists of Bernoulli trials (i.e. each trial has two
possible outcomes)
- probability remains known, and constant throughout the
trials
- number of trials is known
- all trials are random and independent
The described experiment satisfies all of the above, so
binomial distribution is applicable.
Using a pair of FAIR dice, probability of rolling a
double-six is
p=(1/6)^2=1/36.
Number of throws
n=20
Number of times double-sixes are rolled
x=2
Using the standard binomial formula
P(2)=C(n,x)p^x (1-p)^(n-x)
=C(20,2)(1/36)^2(35/36)^(20-2)
=(20*19/(1*2))(1/36)^2*(35/36)^18
=0.0882934
Probability of rolling a double, p = 6(1/36) = 1/6
To calculate the probability of rolling at least two doubles
(out of 20), we will calculate probabilities of rolling zero or one double. Then subtract the sum from 1 to get
probability of rolling 2,3,4,5…20 doubles.
P(0)=C(20,0)p^(0) (1-p)^(20-0)
=1*(1/6)^0*(5/6)^20
=0.026084
P(1)=C(20,1)p^(1) (1-p)^(20-1)
=20*(1/6)(5/6)^(20-1)
=20*(1/6)(5/6)^19
=0.104336
P(2≤x≤20)
=1-P(0)-P(1)
=1-0.025084-0.104336
=0.86958
