If seven flowerbeds are each planted with 20 seedlings find the probability that at least 12 seedlings mature in exactly five of the flowerbeds.
Providing each seedling has a 0.7 chance of developing into a mature plant.
Hello, BabyMilo!
In any one particular flowerbed:A seedling has a 0.7 probability of developing into a mature plant.
Seven flowerbeds are each planted with 20 seedlings.
Find the probability that at least 12 seedlings mature in exactly five of the flowerbeds.
. . $\displaystyle \begin{array}{ccccc}
P(\text{12 mature}) &=& {20\choose12}(0.7)^{12}(0.3)^8 \\ \\[-3mm]
P(\text{13 mature}) &=& {20\choose13}(0.7)^{13}(0.3)^7 \\ \\[-3mm]
P(\text{14 mature}) &=& {20\choose14}(0.7)^{14}(0.3)^6 \\
\vdots && \vdots \\
P(\text{19 mature}) &=& {20\choose19}(0.7)^{19}(0.3)^1 \\ \\[-3mm]
P(\text{20 mature}) &=& {20\choose20}(0.7)^{20}(0.3)^0\end{array}$
The probability that at least 12 mature is the sum of the above probabilities.
. . Call this sum $\displaystyle S.$
We have a choice of 5 of the 7 flowerbeds: .$\displaystyle {7\choose5}$
. . Therefore: .$\displaystyle {7\choose5}\times S$