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In statistics, a random variable is a variable that can take on different values based on the outcome of a random event. In our context, we are looking at the number of rhubarb plants that might die before producing any rhubarb, which is denoted by the random variable \( Y \).
This random variable can theoretically take on values from 0 to 10, as there are 10 plants in total. So if we describe \( Y \):

• \( Y = 0 \): None of the plants die.
• \( Y = 1 \): Exactly one plant dies.
• \( Y = 10 \): All 10 plants die.

The randomness here arises because which plants, if any, actually die is not something we can predict with certainty, hence \( Y \) is random.

A binomial distribution is a probability distribution that summarizes the likelihood of a value coming from a set of independent events, each with the same probability of occurrence. In this case, we have 10 independent trials (plants), each trial having two possible outcomes: dying or surviving.
The distribution is termed 'binomial' because it models scenarios that have precisely two outcomes per trial. Suppose you fix the number of success trials defined by a random variable, and you have a consistent probability of success for each trial. In that case, you are dealing with a binomial distribution—it fits this rhubarb example perfectly.
Mathematically, a binomial distribution can be represented using the parameters \( n \) (number of trials) and \( p \) (probability of success/the event occurring). Using these elements, we determine the probability of a particular number of successes (dead plants in this scenario) from these trials.

The probability of success in a binomial distribution refers to the probability that a single trial results in a 'success'. This can be somewhat counterintuitive, as 'success' is merely the term used for the outcome we are focused on measuring. In our exercise, a plant dying is deemed a 'success' since that's the outcome of interest that we are examining probability calculations for.
For these rhubarb plants, the probability of a plant dying, or success, is 0.05, as 5\% of the plants on average might die. This single probability value \( p = 0.05 \) is essential for calculating probabilities regarding any number of plant deaths or survival. Given this percentage, it allows us to calculate other potential outcomes using the binomial probability formula.

The binomial coefficient is a critical component in calculating binomial probabilities. It is denoted as \( \binom{n}{k} \) in mathematical formulas, and it represents the number of ways to pick \( k \) successes from \( n \) trials.
The binomial coefficient is computed using factorials and can be written as:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Where \( n! \) ("n factorial") is the product of all positive integers up to \( n \). In the rhubarb plant exercise, the binomial coefficient \( \binom{10}{1} \) calculates to 10, representing the 10 different possibilities for one plant to die out of 10.
This component informs us how many sequences exist for achieving exactly \( k \) successes in \( n \) trials, important for understanding the full formula used for probability calculations in binomial distributions.