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To calculate the probability of getting exactly one head in three independent coin flips, the binomial formula can be applied. In this scenario, the number of trials ($n$) is 3, the desired number of successes ($k$) is 1, and the probability of a single success (heads, $p$) is 0.5. Substituting these values into the formula, $\binom{3}{1}(0.5)^1(0.5)^{3-1}$, results in a calculation of $3 \cdot 0.5 \cdot 0.25$, which equals 0.375.

Applying the Binomial Formula for One Head in Three Flips

A machine manufactures electronic chips. Due to a known issue, each chip produced has an independent 50% chance of being faulty. A quality control engineer randomly selects 3 chips for testing. Based on this information, calculate the exact probability that only one of the three selected chips is faulty.

Manufacturing Quality Control

A simple genetic trait is determined by a single gene with two alleles, where the dominant allele has a 50% chance of being passed to an offspring. If a parent passes this gene to three separate offspring, what is the probability that exactly one of the three offspring inherits the dominant allele?

A student is taking a 3-question quiz where each question is a true/false choice. The student guesses randomly on all questions. The probability of guessing exactly one question correctly is calculated using the expression: $3 \times (0.5)^1 \times (0.5)^2$. Explain what each of the three components of this expression (the '3', the '$(0.5)^1$', and the '$(0.5)^2$') represents in the context of this specific scenario.

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