Understanding Binomial Distribution and Transformations

The question asks about the distribution of a new random variable Y, defined as Y = 10 - X, where X follows a Binomial distribution with specific parameters. To solve this, we first need to understand the properties of a Binomial distribution and how transformations affect it.

A random variable X follows a Binomial distribution with parameters n and p, denoted as \( \rm X \sim B(n, p) \), if it represents the number of successes in n independent Bernoulli trials, where the probability of success in each trial is p. The probability of failure in each trial is q = 1 - p.

In this problem, we are given that X follows a Binomial distribution with:

• Number of trials, \( \rm n = 10 \)
• Probability of success in each trial, \( \rm p = \dfrac{1}{5} \)

So, \( \rm X \sim B\left(10, \dfrac{1}{5}\right) \). The probability of failure in each trial is \( \rm q = 1 - p = 1 - \dfrac{1}{5} = \dfrac{4}{5} \).

Analyzing the Transformation Y = 10 - X

The new random variable is defined as \( \rm Y = 10 - X \). Since X represents the number of successes in 10 trials, Y = 10 - X must represent the number of *failures* in the same 10 trials.

Let's consider the outcomes of the 10 independent trials:

• Each trial results in either a success (with probability \( \rm p = 1/5 \)) or a failure (with probability \( \rm q = 4/5 \)).
• The total number of trials is fixed at 10.
• The trials are independent.

If X counts the number of successes out of 10 trials, then Y = 10 - X counts the number of failures out of these same 10 trials. The probability of a 'failure' in the context of X is actually the 'success' probability in the context of Y.

Determining the Parameters for Y

Y represents the number of failures in 10 independent trials. The probability of a failure in a single trial is \( \rm q = 1 - p = \dfrac{4}{5} \).

Therefore, Y also follows a Binomial distribution with:

• Number of trials, \( \rm n' = 10 \) (the total number of trials remains 10)
• Probability of 'success' (which is failure for X) in each trial, \( \rm p' = q = \dfrac{4}{5} \)

So, \( \rm Y \sim B\left(10, \dfrac{4}{5}\right) \).

Comparing this with the general form \( \rm B(n', p') \), we find that \( \rm n' = 10 \) and \( \rm p' = \dfrac{4}{5} \).

Summary of Distribution Parameters

Thus, Y follows a Binomial distribution with parameters n' = 10 and p' = \( \dfrac{4}{5} \).

Revision Table: Key Binomial Concepts

Additional Information: Properties of Binomial Distribution

The Binomial distribution is a fundamental discrete probability distribution. Here are some additional points:

• The probability mass function (PMF) of \( \rm X \sim B(n, p) \) is given by \( \rm P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \) for \( \rm k = 0, 1, \ldots, n \).
• The mean of a Binomial distribution is \( \rm E(X) = np \).
• The variance of a Binomial distribution is \( \rm Var(X) = np(1-p) \).
• The sum of two independent Binomial random variables with the same probability of success \( \rm p \), say \( \rm X_1 \sim B(n_1, p) \) and \( \rm X_2 \sim B(n_2, p) \), follows a Binomial distribution \( \rm X_1 + X_2 \sim B(n_1 + n_2, p) \).
• The transformation \( \rm Y = n - X \) essentially switches the roles of success and failure, hence the probability parameter changes from \( \rm p \) to \( \rm 1-p \). The number of trials remains the same because Y counts outcomes from the same set of n trials.