Unseen Coins A statistics professor tosses two coins that cannot be seen by any students. One student asks this question: “Did one of the coins turn up heads?” Given that the professor’s response is “yes,” find the probability that both coins turned up heads.

Verified step by step guidance

1. Define the sample space for tossing two coins. The possible outcomes are: HH (both heads), HT (head and tail), TH (tail and head), and TT (both tails).
2. Identify the event of interest: 'both coins turned up heads' (HH). This is the event we want to calculate the probability for, given the professor's response.
3. Determine the condition provided in the problem: The professor answered 'yes' to the question 'Did one of the coins turn up heads?' This means the outcomes TT (both tails) are excluded from consideration, leaving the reduced sample space: HH, HT, and TH.
4. Use the formula for conditional probability: P(A|B) = P(A ∩ B) / P(B), where A is the event 'both coins turned up heads' and B is the event 'at least one coin is heads.'
5. Calculate the probabilities: P(A ∩ B) is the probability of HH (since HH satisfies both A and B), and P(B) is the probability of the reduced sample space (HH, HT, TH). Divide P(A ∩ B) by P(B) to find the conditional probability.

Key Concepts

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. In this scenario, we need to calculate the probability of both coins showing heads, given that at least one coin shows heads. This concept is crucial for understanding how the initial information (the professor's 'yes' response) affects the overall probability.

Sample Space

The sample space is the set of all possible outcomes of a random experiment. For the two coins, the sample space consists of four outcomes: HH (both heads), HT (first head, second tail), TH (first tail, second head), and TT (both tails). Understanding the sample space helps in determining the relevant outcomes when calculating probabilities.

Bayes' Theorem

Bayes' Theorem provides a way to update the probability of a hypothesis based on new evidence. In this case, it can be used to revise the probability of both coins being heads after receiving the information that at least one coin is heads. This theorem is essential for making informed conclusions in situations involving conditional probabilities.