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Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1. A probability of 0 means the event will not happen, and a probability of 1 means the event will definitely happen. For any event, finding its probability involves figuring out how often it happens out of all possible outcomes.

In the context of the given exercise, probability helps us determine the likelihood that a certain number of workers use public transportation daily. This is an example of a binomial probability, where we need to calculate the probability of a fixed number of successful outcomes. For our exercise, we figured out the probabilities of exactly three and at least three workers using public transportation with the use of the binomial probability formula.

• Use decimal format to express probabilities, like 0.3 for 30%.
• The sum of all probabilities for a scenario’s possible outcomes equals 1.
• Fractions or percentages are common ways to express probabilities.

A random variable is a mathematical variable that represents a possible outcome of a random event. It can take different values based on the probability of occurrences in an experiment or scenario.

In our exercise, the random variable "X" was defined to represent how many out of the 10 workers take public transportation daily. "X" can take any integer value from 0 to 10. Each of these values is a possible number of workers taking public transport, and each has a corresponding probability which can be calculated.

• Random variables can be discrete or continuous. "X" is a discrete random variable since it only takes whole numbers.
• The expected value and variance are important characteristics of random variables.
• Understanding random variables is key in probabilities, as they help quantify variability in different outcomes.

The binomial coefficient, often represented as \( \binom{n}{k} \), is a central part of the binomial distribution. It illustrates the number of ways to choose "k" successes in "n" trials. This coefficient multiplies with probabilities in the binomial formula.

For example, in our scenario, the binomial coefficient \( \binom{10}{3} \) equals 120. This means there are 120 different ways in which exactly 3 workers could choose to take public transportation out of 10 workers. The formula for calculating a binomial coefficient is: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Where "!" denotes factorial, meaning you multiply a series of descending natural numbers. E.g., \( 10! = 10 \times 9 \times 8 \times \ldots \times 1 \).

• Binomial coefficients are symmetrical: \( \binom{n}{k} = \binom{n}{n-k} \).
• This concept is important in combinatorics and probability.
• In calculating binomial probabilities, each possible outcome is recombined differently, as expressed by the coefficient.