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In this exercise, calculating the probability of different outcomes for defective boards involves understanding and computing various probabilities. The primary tool used here is the binomial probability formula, which helps us calculate the likelihood of obtaining exactly a certain number of defects, as each trial (testing a board) results in either a defective or non-defective board. We want to find the probability that at most 2 boards, at least 5 boards, and between 1 to 4 boards are defective. Also, calculate the probability of having no defective boards.

• For probabilities like \(P(X \leq 2)\), we find the sum of probabilities for 0, 1, and 2 using the formula: \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\).
  
• When calculating \(P(X \geq 5)\), we find the complement \(P(X \leq 4)\) and subtract it from 1.
  
• To obtain probabilities like \(P(1 \leq X \leq 4)\), calculate each probability from 1 to 4 and add them up, which illustrates how likely it is to see these outcomes.
  

This method allows for precise determination of probabilities in a binomial distribution setup, highlighting the calculation of possibilities using the defined success (defective board) rate.

The expected value in a binomial distribution gives us a snapshot of the average or mean number of successes across many trials. It is a vital measure that helps understand what to anticipate when repeating an experiment many times.
To compute the expected value for our scenario, use the formula:
\[E(X) = np\]
Here, \(n\) is the total number of trials (25 boards), and \(p\) is the probability of one trial being a success (5% defect rate).

• This results in an expected number of defective boards as \( E(X) = 25 \times 0.05 \), yielding 1.25 defective boards.
  

The expected value provides a clear indication of the number of defects you might "expect" on average every time you test 25 boards, though actual outcomes can vary around this average due to random variation.

Standard deviation is a critical concept in statistics that quantifies the amount of variation or dispersion in a set of values in a distribution.
For the binomial distribution in this context, the formula for standard deviation is:
\[\sigma = \sqrt{np(1-p)}\]
This measure helps you understand how much actual results can deviate from the expected value due to variability in the data.

• Plugging in our values (\(n = 25\), \(p = 0.05\)), we calculate \( \sigma = \sqrt{25 \times 0.05 \times 0.95} \).
  
• This yields a standard deviation of approximately 1.089.
  

A larger standard deviation indicates more variation from the expected number of defects. In our case, this helps in assessing risk and variability, assisting in making informed decisions based on the data's spread.

The binomial probability formula is the backbone of calculating probabilities in scenarios where you have a fixed number of independent experiments, each with the same probability of success. It is given by:
\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
Here's how it breaks down:

• \(n\) is the number of trials.
  
• \(k\) is the number of successful trials (defective boards, in our case).
  
• \(p\) is the probability of success in a single trial.
  
• \((1-p)\) is the probability of failure.
  
• \(\binom{n}{k}\) is a combination function calculating the different ways \(k\) successes can occur in \(n\) trials.
  

This formula helps in evaluating probabilities for any specific number of successes in the context of binomial distribution, such as testing circuit boards in our example. Mastering this formula allows for versatile application across various real-world scenarios where similar conditions of fixed trials and constant probabilities apply.