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This example demonstrates how to calculate the probability of a specific outcome from multiple independent events. Consider a scenario where three friends each flip a fair coin once, and the question is: what is the probability that exactly one of them flips heads? 

To solve this, we first identify the sample space, which consists of all possible outcomes. Since each of the three flips has two outcomes (Heads or Tails), there are $2^3 = 8$ total possibilities: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. 

Next, we identify the favorable outcomes that match the condition of having exactly one head: HTT, THT, and TTH. There are 3 such outcomes. 

The probability is the ratio of favorable outcomes to the total number of outcomes, which is 3/8.

Calculating the Probability of Exactly One Head in Three Coin Flips

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 student is trying to calculate the probability of getting exactly two heads from three fair coin flips. Their reasoning is as follows: 'There are 8 total possible outcomes (2 outcomes per coin for 3 coins, so 2x2x2=8). For the favorable outcome, we need two heads and one tail, such as HHT. Therefore, there is only one favorable outcome. The probability is 1/8.' Which statement best analyzes the error in this reasoning?

Given the quality control scenario described below, calculate the probability that a randomly selected sample of three components will pass inspection. Explain your reasoning.

Quality Control Probability

The probability of obtaining exactly one 'Heads' in three fair coin flips is identical to the probability of obtaining exactly two 'Heads' in three fair coin flips.

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