Question

N=20 p=0.3. Using the probability notation P(x=x), write out all the probabilities you'd need to add up in order to calc P(x<6) then evaluate using a calculator.

Answer 1

`\mathbb P(X<6)=\mathbb P(X=0)+\mathbb P(X=1)+\cdots+\mathbb P(X=5)`


`X` appears to have a binomial distribution, which means


`\mathbb P(X=x)=\dbinom Nx p^x(1-p)^(N-x)`

where
`\dbinom nk=(n!)/(k!(n-k)!)`. So, as an example, you could compute


`\mathbb P(X=0)=\dbinom{20}00.3^0*0.7^(20)=0.7^(20)\approx0.0007979`

You should get


`\mathbb P(X<6)=\displaystyle\sum_(x=0)^5\mathbb P(X=x)\approx0.4162`