Question

It was found that 90% of cucumber seeds sown in the soil germinate. Determine the most likely

number of germinated grains if there are 70 grains in the package.

Answer 1

By the binomial distribution, the probability that k seeds germinate is $\begin{pmatrix} 70 \\ k \end{pmatrix}(0.9)^k(1-0.9)^{70-k}$

The expected value of the number of seeds to germinate is $pn=70*0.9=63$.

So, we need to compute the binomial distribution probability values for k=62, k=63 and k=64 to find the maximum.

For $k=62$

$P=\frac {70!} {(70-62)!62!} (0.9)^{62}(0.1)^{70-62}=0.1374$

For $k=63$

$P=\frac {70!} {(70-63)!63!} (0.9)^{63}(0.1)^{70-63}=0.15704$

For $k=64$

$P=\frac {70!} {(70-64)!64!} (0.9)^{64}(0.1)^{70-64}=0.15459$

As we can see, $P(k=63)=0.15704$ is the most probable.

So, the most likely number of germinated grains if there are 70 grains in the package is 63.