The statement "Half of the 23 students in a class have the same birthday" can be described as unlikely.
Explanation of Likelihood:
The birthday problem (or birthday paradox) suggests that in a group of 23 people, there is about a 50% chance that at least two people share a birthday. However, the probability of exactly half of the students (i.e., 12 of the 23) all sharing the same birthday is extremely low due to the distribution of birthdays across 365 days in a year. Statistically, the likelihood of such a scenario occurring decreases as the required number of individuals sharing a birthday increases.
Similar Statements with Different Likelihoods:
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Statement 1: "At least two students in the class share the same birthday."
- Likelihood: Likely.
- Reasoning: In a group of just 23 students, there is a well-documented probability that at least two individuals will share a birthday, estimated at around 50%.
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Statement 2: "All 23 students in the class have unique birthdays."
- Likelihood: Possible but unlikely.
- Reasoning: While it is theoretically possible for all students to have unique birthdays, the probability is significantly lower than that of having at least one shared birthday. The probability decreases rapidly as more individuals are added to the group.
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Statement 3: "Exactly 3 students in the class share the same birthday."
- Likelihood: Unlikely.
- Reasoning: The probability of exactly three students sharing a specific birthday while the rest do not follow a similar distribution is also quite low, although slightly more plausible than half of them sharing a birthday.
By analyzing the statistical properties of the birthday problem and probabilities related to shared birthdays, we can confidently assess the likelihoods of these statements in the context provided.