Question

Dan is planting 24 iries in his front yard.The flowers he bought were a combination of two varieties ,blue and white .The flowers are not blooming yet ,but Dan knows that the probability of having a blue flower is 75%
1)what is probability that at least 20 of white irises in Dan's garden?
2)What is rhe expected mumber of white irises in Dan's garden ?

Answer 1

Probability of atleast 20 white irises = Probability of 4 or less Blue
Prob(#Blue=x)=\[\left(\begin{matrix}24 \\ x\end{matrix}\right)\]times (.75)^x*(.25)^(24-x)
Now sum this for x=0, 1, 2, 3, 4 and thats the answer to the first question.
Note:the thing in parenthesis is 24 choose x. This is also written as 24!/(x!*(24-x)!)

Expected value of #of White irises = \[\sum_{x=0}^{24} x*P(number white=x)\]
P(numberwhite=x) = (24 choose x)*(.25)^x*(.75)^(24-x)

There is a theorem that since each flower is independent of eachother, then you just take the expected value (white=1 and blue=0) for each flower and sum them. Since each flower has .25 prob. to be white. It is just 24*.25= 6 expected white flowers