The probability that a car will not develop a major fault within the first 3 years of its life is \(0.997\). Calculate the probability that of 20 cars selected at random (a) 19 will not develop any major faults in the first 3 years (b) 19 or more will not develop any major faults in the first 3 years.

The binomial probability formula is essential for solving problems that ask for the probability of having a specific number of successes in a set of independent trials. In our exercise, the success is defined as a car not developing a major fault within the first three years.

The formula is written as:
\[ P(X=k) = C(n,k) \times p^k \times q^{(n-k)} \]
Where:

• \( P(X=k) \) is the probability of getting exactly \( k \) successes in \( n \) trials.
• \( C(n,k) \) is the number of combinations of \( n \) things taken \( k \) at a time.
• \( p \) is the probability of success on any given trial.
• \( q \) is the probability of failure on any given trial (\( q = 1 - p \)).

This is a fundamental tool in statistics that helps evaluate the likelihood of varied outcomes where there are distinct 'success' or 'failure' results.

In our exercise for part (a), the binomial probability formula was applied to calculate the probability that 19 out of 20 cars would not develop any major faults within the first three years. Such problems are common in quality control processes, reliability testing, and predictive modeling, making the binomial probability formula a versatile and powerful tool.

The concept of combinations is central to calculating probabilities in scenarios where the order of outcomes does not matter. It refers to selecting items from a larger set such that the arrangement of these items is irrelevant.

Mathematically, the number of ways to choose \( k \) elements from a set of \( n \) distinct elements is shown using the combination formula:
\[ C(n,k) = \frac{n!}{k!(n-k)!} \]
Here, \( n! \) (n factorial) is the product of all positive integers up to \( n \), and \( k! \) is the product of all positive integers up to \( k \). The combination formula helps us determine how many different groups of a certain size can be made from a larger pool, without considering different orders within a group.

To connect this with our car reliability example, the combination\( C(20,19) \) calculates the number of ways we can select 19 cars out of 20 that will not develop major faults.

The concept of factorial calculation plays a critical role in both binomial probability and combinations. It is denoted by an exclamation mark (!) and it means multiplying a series of descending natural numbers.

Specifically,
\[ n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1 \]
where \( n \) must be a non-negative integer, and by convention, \( 0! = 1 \).

In probability and combinatorics, factorial calculations allow us to find out the total number of possible outcomes. For instance, in the car problem, calculating \( 20! \) gives the total number of ways to arrange 20 cars, which is key to then finding out how many of those arrangements meet our specific 'no faults' criteria.

Understanding factorial calculation is crucial because it is the foundation upon which combinations and other permutations are based. Whether you're looking at a line of people, a hand of cards, or a sequence of events, factorial calculation helps quantify the different ways in which something can occur or be arranged.