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The binomial distribution is a fundamental concept in probability theory. It is used when you have a fixed number of independent trials, such as flipping a coin or making sales calls, each with the same probability of success.
For this exercise, the salesperson making contacts with prospects corresponds to the trials, and making a sale is the success event. The key parameters you'll need to know are:

• The number of trials (n), which is 100 contacts in this example.
• The probability of success (p) on a single trial, given as 0.03.

The binomial distribution can answer questions about how likely it is to achieve a given number of successes in the specified number of trials. This problem centers on finding the probability of making at least one sale (at least one success) in 100 trials.

Complementary probability is a handy concept for solving problems where you're interested in the "at least one" type of outcome. When calculating the probability of at least one sale, it's often easier to find the probability of its complement, which is no sales, and then subtract from 1.
If you know the probability of an event occurring, the complementary probability helps you find the probability of it not occurring. Here's the basic formula:

• For an event with probability p, the complementary probability is 1-p.

In the context of our exercise, the complementary probability is used to determine the likelihood of making no sales (0.97) on a single contact, helping us step closer to finding the probability of at least one sale.

Calculating the probability of no event, which in this case is no sales being made, is a crucial intermediate step. To find the probability that no sales occur across 100 contacts, you multiply the complementary probability of not making a sale on one contact by itself 100 times. Mathematically, this is expressed as:

• \[ (1-p)^n = 0.97^{100} \]

The result tells us the chance that the salesperson will not make any sales at all in 100 attempts. This step allows you to then find out the likelihood of making at least one sale by using complementary probability once more.

Calculating probabilities step by step ensures you don't miss any critical pieces of the solution puzzle. Here’s how you can calculate the probability of making at least one sale:
1. **Determine Probabilities**: Identify that each contact has a 0.03 chance of a sale.
2. **Complementary Event**: Calculate that the chance of no sale on any single contact is 0.97.
3. **No Sales in n Trials**: Use power of complementary probability: \(0.97^{100}\).
4. **At Least One Sale**: Subtract probability of no sales from 1: \(1 - 0.97^{100}\).
5. **Calculate Final Probability**: Solve equations using a calculator to find \(1 - 0.97^{100} \approx 0.95\).
Following these steps results in the solution to the problem, showing that there's a 95% chance of making at least one sale in 100 contacts.