A Poisson distribution models voter arrivals at a polling station with an average rate of 5 voters per 20 minutes. Calculate: a) The probability of no voters arriving between 3:10 pm and 3:30 pm. b) T

The time interval is from 3:10 pm to 3:30 pm, which is $3:30 - 3:10 = 20$ minutes.
For a 20-minute interval, the average rate $\lambda = 5$.

We want to find $P(X=0)$.

The probability that no voter will arrive is $\boxed{\text{0.006738}}$.

The time interval is from 9:20 to 10:20, which is $10:20 - 9:20 = 60$ minutes.
Since the rate is 5 voters per 20 minutes, for a 60-minute interval, the new average rate $\lambda'$ is:

We want to find $P(X=5)$.

The probability that there will be exactly 5 voters is $\boxed{\text{0.001936}}$.

For a Geometric distribution, the mean $E(X)$ is given by:

For a Geometric distribution, the variance $Var(X)$ is given by:

The mean is $\boxed{\text{5}}$ and the variance is $\boxed{\text{20}}$.

We want to find $P(X=5)$.

The probability that he finds a wife on the 5th church service is $\boxed{\text{0.08192}}$.

For a Geometric distribution, $P(X > n) = (1-p)^n$.
We are given $P(X > n) < 1\%$, which is $P(X > n) < 0.01$.
So, we need to solve:

Take the logarithm (base 10 or natural log) of both sides. Using $\log_{10}$:

Calculate the logarithm values:
$\log_{10}(0.8) \approx -0.09691$
$\log_{10}(0.01) = -2$
Substitute these values into the inequality:

Divide by $-0.09691$. Remember to reverse the inequality sign when dividing by a negative number:

Since $n$ must be an integer and $n > 20.637$, the least integer value for $n$ is 21.
The least integer $n$ is $\boxed{\text{21}}$.