Question

let Y1, Y2, ..., Y100 are Bernoulli variables with a value of 1 and probability 0.5. Estimate the probability that Y1 + Y2 + ... + Y100 < 60.

assume all variables are independent.

Answer 1

If Y_i's ; i = 1 , 2, 3, ..., n are independent Bernoulli variables with probability of success is p , then their sum follows Binomial random variable with parameter n and probability of success = p.

Therefore here Y = Y₁ + Y₂ ...+ Y₁₀₀ follows Binomial distribution with parameters n = 100 and p = 0.50

We want to find P( Y < 60 ) = P( Y <= 60 - 1 ) = P(Y <= 59)

Let's use excel:

P( X <= 59) = "=BINOMDIST(59,100,0.5,1)" = 0.971556   (This is the final answer).