For a binomial experiment, show that the probability of

Question

For a binomial experiment, show that the probability of $x$ successes $P(x)$ is given by:

$P(x) = \binom{n}{x} p^x (1 - p)^{n-x}.$

11 Jan 2025

Answer :

Word Count : 1176

In a binomial experiment, we deal with a situation where each trial has two possible outcomes: success or failure. A typical example is flipping a coin where the outcomes can be "heads" (success) or "tails" (failure). The binomial experiment is characterized by the following properties:

1. There are $ n $ independent trials.
2. Each trial has two possible outcomes: success (denoted as $ x $) or failure (denoted as $ n - x $).
3. The probability of success on any given trial is $ p $, and the probability of failure is $ 1 - p $.
4. The number of successes, $ x $, in $ n $ trials is of interest.

### Step 1: Understanding the Problem

We are interested in finding the probability of having exactly $ x $ successes in a binomial experiment. The total number of trials is $ n $, and the probability of success in each trial is $ p $, while the probability of failure is $ 1 - p $.

### Step 2: Defining the Binomial Probability

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