Calculating the Probability of Exactly One Head in Three Coin Flips
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.
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Ch.3 Prompting - Foundations of Large Language Models
Foundations of Large Language Models
Computing Sciences
Foundations of Large Language Models Course
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Calculating the Probability of Exactly One Head in Three Coin Flips
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Calculating the Probability of Exactly One Head in Three Coin Flips
An AI model is prompted with the following word problem: 'Three friends play a game where they flip a fair coin. Each friend flips the coin once. What is the probability that exactly one of them flips heads?'
Below are two different initial responses generated by the AI. Analyze the strategies and determine which response represents a more effective approach for solving this type of multi-step reasoning problem.
Response A: "The probability of one person getting heads is 1/2. Since there are three people, and we want exactly one to get heads, the probability is 1/3 multiplied by 1/2, which equals 1/6."
Response B: "Let’s think step by step. First, let's list all the possible outcomes for the three coin flips. Second, let's identify which of those outcomes have exactly one head. Third, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes."
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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?
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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.