Probability of Gold Flakes in a Sample

Suppose a beverage manufacturer adds gold flakes into a beverage at a rate of 800 flakes per litre. Given a 10 millilitre sample taken from a well-shaken 1 litre bottle, what is the probability that the sample contains exactly 2 flakes of gold?

This problem involves discrete probability, a key concept in understanding how likely events are to occur in a finite sample space. Probability theory is the backbone of many areas of mathematics and computer science. In this particular case, we need to determine the probability that a sample from a larger population contains a specific number of items—in this case, gold flakes.

To solve this problem, we will likely apply the concepts of binomial probability distribution and the binomial coefficient. The binomial distribution is a common discrete probability distribution that models the number of successes in a fixed number of independent experiments, each with the same probability of success. Here, taking a sample from a well-shaken bottle is akin to conducting a series of Bernoulli trials.

This type of problem is useful in various fields, including quality control, where manufacturers might want to determine the likelihood of producing a certain number of defective items in a batch, or in computer science when working with algorithms that involve random sampling. Understanding how to compute such probabilities not only enhances your problem-solving skills but also prepares you for tackling more complex stochastic processes and statistical models.