Answer:
The answer is 46/512 = 0.08984375
Explanation:
In this question, success means to get an odd number on rolling a die.
The probability of success p = 3/6 = 1/2
So p = 1/2
Now lets find probability of failure q
q = 1 - p = 1 - 1/2 = 1/2
So q = 1/2
Let X = x denotes the number of success in n trials.
So X is a Binomial Random Variable
Which has the following parameters:
n = 9
as the fair die is rolled nine times and
p = 1/2
Now the Probability of x success out of n trials P( X = x) is:
P(X = x) = p(x) = nCx pˣ , qⁿ⁻ˣ , x= 0,1,2,...,9
P(X = x) = p(x) = nCx (1/2)⁹ = ₉Cₓ (1/2)⁹ = ₉Cₓ /512
Since the required probability is P (X < 3) So
P(X < 3) = P(X = 0) + P( X = 1) + P(X = 2)
= 1 / 512 {₉C₀ + ₉C₁ + ₉C₂} nCr = n! / r! * (n - r)!
= 1 / 512{ (9! / 0! * (9 - 0)!) + (9! / 1! * (9 - 1)!) + (9! / 2! * (9 - 2)!) }
= 1 / 512 { 1 + 9 + 36 }
= 46 / 512 = 0.08984375
So the required probability is 46/512