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In probability and statistics, understanding conditional distribution is crucial as it describes the probability of an event given that another event has occurred. For this particular problem, we're interested in the conditional distribution of the individual test results of each vehicle, given the total number of vehicles that passed the test.

In the context of our exercise, we define the conditional distribution as:

• We consider three vehicles, each either passing (1) or failing (0) the emissions test.
• Given that exactly \( x \) vehicles pass, we need to find the probability distribution of possible outcomes resulting in \( x \) total passes.

Given the total passes \( X = x \), the conditional distribution for the individual outcomes \( X_1, X_2, \) and \( X_3 \) becomes a crucial factor that indicates how the passes are distributed among the vehicles.

Since the conditional probability doesn't depend on \( p \), the outcomes are uniformly likely, shown by the probability \( P(X_1 = x_1, X_2 = x_2, X_3 = x_3 \mid X = x) = \frac{1}{\binom{3}{x}} \). This uniformity in distribution, regardless of the value of \( p \), shows that the total number of passes is a sufficient statistic for \( p \).

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states across a number of trials. In this exercise, each vehicle's test is a trial, and its outcome is a success (passing) or a failure (failing).

The sum \( X = X_1 + X_2 + X_3 \) represents a binomial random variable because it is the result of adding up independent Bernoulli random variables (each with the same probability \( p \) of passing). For 3 vehicles, \( X \) follows a binomial distribution with parameters 3 (the number of trials) and \( p \) (the probability of success in each trial). Formally, it is expressed as:

\[ X \sim \text{Binomial}(3, p) \]

This distribution tells us the probabilities of getting 0, 1, 2, or all 3 vehicles passing the test. It plays a crucial role in determining the sufficiency of the statistic \( X \) because, by summing up results from independent tests, we're effectively compressing all information about \( p \) into a single statistic.

A Bernoulli random variable is a specific type of random variable that has only two possible outcomes, usually labeled as 1 (success) and 0 (failure). In our vehicle emission test scenario, each vehicle's result can either pass or fail, making it a Bernoulli random variable.

Every \( X_i \) in the original problem (where \( i = 1, 2, 3 \)) is a Bernoulli random variable with a probability \( p \) of passing the test. The probability mass function is given by:

\[ P(X_i = x_i) = p^{x_i}(1-p)^{1-x_i} \]

where \( x_i \) is either 0 or 1. This concise representation highlights the binary nature of outcomes for each test.

• Understanding individual outcomes helps in identifying how these variables can be added together to form a Binomial distribution.
• The behavior of the Bernoulli variables (their likelihood of success and independence) plays a central role in assessing the sufficiency of the sum \( X = X_1 + X_2 + X_3 \). When combined, they transition into a binomial model which aids in testing the sufficiency for the parameter \( p \).