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Probability calculation is an essential tool in determining the likelihood of an event happening. In the context of our exercise, we need to calculate the probability that exactly two out of ten telephones need replacement under warranty. This requires understanding both the overall probability of individual occurrences and how these contribute to the larger event.
We start by finding the probability of a single telephone being submitted and subsequently being replaced. Since 20% of telephones are submitted and only 40% of those need replacement, we calculate this by multiplying these probabilities:

• The probability of a telephone being submitted is 0.20
• The probability of a submitted phone being replaced is 0.40
• The combined probability for a phone being replaced is therefore: \(0.20 \times 0.40 = 0.08\)

Understanding this calculation helps us build toward the overall probability for multiple phones, using this base probability that a single phone will be replaced.

The binomial probability formula is a powerful method for calculating the probability of a specific number of successes in a series of independent and identical trials. In our case, each telephone's need for replacement is considered an independent trial.
The formula for the probability of getting exactly \(k\) replacements out of \(n\) phones is:

• \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)

Here,

• \(n\) represents the total number of trials, which is 10 phones
• \(k\) is the number of successes we want, which is 2 replacements
• \(p\) is the probability of a single success, which is a phone being replaced (0.08)
• \(1-p\) is the probability of a phone not being replaced (0.92)

To compute the exact probability, apply these values into the formula, resulting in:

• \(\binom{10}{2} (0.08)^2 (0.92)^8\)

Understanding each component ensures accurate probability calculations in various situations.

Independent trials are a fundamental concept in binomial distribution, indicating that the outcome of one trial does not impact the outcomes of another. Each trial in our phone replacement scenario is the decision whether a single phone needs replacement.
There are certain characteristics that define independent trials in binomial distributions:

• Each trial has two possible outcomes: success (replacement) or failure (no replacement).
• The probability of a single event (phone needing replacement) remains constant across all trials.
• The trials themselves do not influence each other, meaning the replacement decision of one phone does not affect another.

Understanding these characteristics ensures you can accurately model and predict outcomes using binomial distribution methods—and that is exactly how we confidently determine the probability of having exactly two phones needing replacement from our batch of ten.