Can someone please help me with the this math problem:
If a seed is planted, it has a 70% chance of growing into a healthy plant.
If 8 seeds are planted, what is the probability that exactly 1 doesn't grow?
Writing "G" for "grows" and "N" for "does not grow", this can happen as
NGGGGGGG
GNGGGGGG
GGNGGGGG
GGGNGGGG
GGGGNGGG
GGGGGNGG
GGGGGGNG
GGGGGGGN
In other words, there are 8 possible orders. That is reflected in Romsek's $\displaystyle \begin{pmatrix} 8 \\ 7\end{pmatrix}= \frac{8!}{7!1!}= 8$. The probability of "NGGGGGGG" is, of course, $\displaystyle (.3)(.7)^7= 0.02471$, approximately, and it is easy to see that the probability of "GNGGGGGG", etc is exactly the same- you have the same numbers, just in a different order. So the probability of all 8 orders is 8(0.02471)= 0.19768.