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Shared Birthdays Find the probability that of 25 randomly selected people, at least 2 share the same birthday.
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Ch. 4 - Probability
Triola14th EditionElementary StatisticsISBN: 9780137366446
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TOC.tableOfContents
- Ch. 1 - Introduction to Statistics88
- Ch. 2 - Exploring Data with Tables and Graphs70
- Ch. 3 - Describing, Exploring, and Comparing Data66
- Ch. 4 - Probability112
- Ch. 5 - Discrete Probability Distributions109
- Ch. 6 - Normal Probability Distributions122
- Ch. 7 - Estimating Parameters and Determining Sample Sizes107
- Ch. 8 - Hypothesis Testing75
- Ch. 9 - Inferences from Two Samples94
- Ch. 10 - Correlation and Regression80
- Ch. 11 - Goodness-of-Fit and Contingency Tables29
- Ch. 12 - Analysis of Variance42
Triola14th EditionElementary StatisticsISBN: 9780137366446Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 4.3.26
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.
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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).
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.
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.
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.'
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.
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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.
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5:37
Introduction to 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.
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05:11Sampling Distribution of Sample Proportion
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.
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