This problem involves finding the cumulative probabilities for a binomial distribution. We are given the following information:

• The probability that a voter prefers Candidate A is $\pi = 0.645$.
• The number of voters polled is $n = 5$.

We need to calculate the cumulative probability distribution $P(X < k)$, where $X$ is the number of voters who prefer Candidate A. $X$ follows a binomial distribution:

$X \sim \text{Binomial}(n = 5, \pi = 0.645)$

The probability mass function (PMF) for a binomial distribution is:

$P(X = k) = \binom{n}{k} \pi^k (1 - \pi)^{n - k}$

The cumulative probability $P(X < k)$ is the sum of probabilities from $X = 0$ to $X = k-1$, i.e.,

$P(X < k) = \sum_{i=0}^{k-1} P(X = i)$

Step-by-step Calculation

We will compute $P(X < k)$ for $k = 0$ through $k = 5$ using the binomial distribution formula.

1. Calculate $P(X = k)$ for each $k$:

• $P(X = 0)$
• $P(X = 1)$
• $P(X = 2)$
• $P(X = 3)$
• $P(X = 4)$
• $P(X = 5)$

2. Compute cumulative probabilities