Question A die is thrown 20 times and a success is defined as a one or a six on the upface of the die.
Use the binomial distribution to determine the probability of getting exactly five successes in the 20 trials.
Answer what are the chances of getting a one/six ? 2 out of 6 ways.
So probability of success = p(E) = 2/6
probability of failure = 1-p(E) = 4/6 = P(F), say.
In how many ways can we get 5 successes in 20 trials
E E E E E and then 15 Fs
E E E E 15 Fs and an E
and many other combinations
For each of these ways, the chance is
P(E) * P(E) * P(E) * P(E) * P(E) * P(F)to the power 15, i.e.
P(E)to the power 5 * P(F)to the power 15
But how many combinations are there?
Imagine a line with 20 spots, choose 5 to place E
there are 20 C 5 ways of doing that ( 20!/15! 5!)
So, the overall chances are:
( 20!/15! 5!) * P(E)to the power 5 * P(F)to the power 15!
