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Probability is a fundamental concept in statistics that helps us measure the likelihood of different outcomes. In a binomial distribution, probability refers to the chance of a particular outcome happening in a set number of trials. For example, if we're interested in whether credit card holders default or not, probability lets us quantify that event's likelihood.
In this example, we're given a 7% probability (0.07) that a cardholder will default. Using probability rules, such as for calculating the likelihood there's zero defaulting, involves the binomial probability formula. Here, probability becomes our tool for predicting events under uncertainty and involves concepts like probability mass function and cumulative probability.

Expected value is a key concept that provides a measure of the center of a probability distribution and is synonymous with the mean of the distribution. For a binomial distribution, the expected value is calculated as the number of trials multiplied by the probability of success.
In our exercise, we're curious about how many cardholders out of 12 new applicants we'd expect to default. We use the formula: \[ E(X) = n \cdot p \]where \( n = 12 \) and \( p = 0.07 \). This gives an expected value of 0.84, indicating we expect 0.84 cardholders to default. Expected value helps us manage expectations and make forecasts in decision-making scenarios.

Standard deviation is an important measure that tells us how much individual values in a dataset vary from the expected value or mean. In a binomial distribution, it helps us understand the variability and uncertainty in the number of successful outcomes.
To find the standard deviation, we use the formula:\[ \sigma = \sqrt{n \cdot p \cdot (1-p)} \]Plugging our numbers in (\[ n = 12, p = 0.07 \], and \[ 1-p = 0.93 \]), we calculate the standard deviation to be approximately 0.7986. This illustrates our expected outcomes' variability and helps assess the consistency of data from our predictions.

The complement rule is a useful probability principle that helps calculate the likelihood of an event not happening by instead determining the probability of its complement – the event happening.
For instance, if we want to know the probability that at least one of the cardholders defaults, we first find the probability of none defaulting and subtract it from 1:
\[ P(X \geq 1) = 1 - P(X = 0) \]In our case, the probability of no defaults is approximately 0.4116, so the probability that at least one defaults is 0.5884. The complement rule simplifies probability problems by offering a different approach to solve them.