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Probability is a fundamental concept in statistics that quantifies the likelihood of an event occurring. In the problem provided, probability helps us evaluate whether the newspaper report that 50 people were interviewed before the first refusal is credible or not. The telephone poll has a 'refusal rate' which means there is a fixed likelihood, 20%, that a call will be refused. It's crucial to understand that this is not about predicting a single occurrence, but instead assessing the likelihood of many occurrences together to form a pattern.

For this scenario, the key probability to be calculated is whether 50 people could be interviewed without refusal. The geometric distribution — a tool to calculate the probability distribution of the number of Bernoulli trials (yes/no outcomes) needed for a success (refusal in this case) — indicates that the more trials you conduct, the lower the probability of not encountering a refusal. Therefore, by calculating the complement probability, we understand that the chance of interviewing 50 people before a single refusal should be exceptionally low. This imminence of a refusal becomes more likely with each additional trial, which inversely makes such a high number of successful interviews (without a refusal) exceedingly unlikely.

In statistical terms, the expectation of a random variable represents the mean value or the long-term average that the variable takes on upon repeated trials. For the geometric distribution, the expected number indicates on average how many trials are needed for a single success to occur.

In our exercise, we treat 'refusal' as the 'success' since we're interested in knowing how many calls it takes before we get a refusal. With a success probability (\( p \)) of 20%, the expected number of trials before success is mathematically described as the reciprocal of that probability, which is calculated as \(1/p \). Therefore, the expected number of people a pollster would need to interview before getting a refusal is \(1/0.2 = 5\) people. It's crucial to note that this is an average value; in practice, a refusal could occur before or after five calls but over time, with enough data, the mean will trend towards this expectation.

The reason we're able to use a geometric distribution in our scenario stems from the assumption of independent trials. This means the outcome of one trial does not influence the outcome of another. In the context of our telephone poll, this assumption translates to the probability of getting a refusal being the same each time a new call is made, regardless of results from previous calls.

For instance, after nine refusals, the probability of refusal on the tenth call remains at 20%, not influenced by the outcomes of earlier calls. This concept of independent trials is essential when dealing with probabilities, as the absence of it can alter the probability distribution and thus the expectations and predictions made for the scenario.