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The article "Birth Beats Long Odds for Leap Year Mom, Baby" (San Luis Obispo Tribune, March 2, 1996) reported that a leap year baby (someone born on February 29 ) became a leap year mom when she gave birth to a baby on February \(29,1996 .\) The article stated that a hospital spokesperson said that the probability of a leap year baby giving birth on her birthday was one in \(2.1\) million (approximately .00000047). a. In computing the given probability, the hospital spokesperson used the fact that a leap day occurs only once in 1461 days. Write a few sentences explaining how the hospital spokesperson computed the stated probability. b. To compute the stated probability, the hospital spokesperson had to assume that the birth was equally likely to occur on any of the 1461 days in a four- year period. Do you think that this is a reasonable assumption? Explain. c. Based on your answer to Part (b), do you think that the probability given by the hospital spokesperson is too small, about right, or too large? Explain.

Step by step solution

01

Understanding the Probability Calculation

As stated in the exercise, a leap day occurs once in 1461 days (which are the total days in 4 years). Therefore, the chance of a person being born on a leap day, and then giving birth on a leap day would be \(1/1461 * 1/1461 = 1/2.134*10^6 = 0.0000004683\). This is approximately equal to \(0.00000047\), as stated by the hospital spokesperson. Hence, the spokesperson is merely multiplying the probabilities of the two independent events.

02

Evaluating the Assumption

The assumption made by the hospital spokesperson is that a birth is equally likely to occur on any of the 1461 days in a four-year period. Realistically, this might not be completely accurate since birth rates can vary due to factors such as season, weather, and cultural practices among others. However, without specific birth data to suggest otherwise and for simplicity, it can be seen as a reasonable assumption.

03

Evaluating the Provided Probability

The calculated probability already assumes that births are equally distributed throughout the year. Given our discussion in Step 2, if we consider variations in birth rates due to factors like seasonality, the probability could be slightly higher or lower. However, without specific data, we cannot make a definitive statement and thus the provided probability seems roughly correct given the used assumption.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Leap Year

Leap years are unique years occurring every four years in the Gregorian calendar. They include an extra day—February 29th—to help synchronize the calendar year with the solar year. This adjustment compensates for the fact that the Earth doesn't orbit the sun in exactly 365 days. Instead, it takes approximately 365.25 days to complete one cycle. To account for this discrepancy, an extra day is added every four years.

In leap years, the occurrence of February 29th, or "Leap Day," becomes a statistical curiosity. Because leap years recur every four years, any specific event or occurrence tied to this day—such as a birth—is considerably rarer than events happening on any regular day that appears annually.

Independent Events

In probability theory, two events are considered independent if the occurrence of one does not affect the occurrence of the other. This means the probability of both events happening is the product of their individual probabilities. Mathematically, if event A and event B are independent, the probability of both A and B occurring is denoted as:

• \[ P(A \cap B) = P(A) \times P(B) \]

In the context of the leap year exercise, giving birth on February 29th is considered independent from being born on that date. Hence, the probability of a leap year baby giving birth on their own birthday is determined by multiplying the probability of each separate event.

Uniform Distribution

A uniform distribution assigns equal probability to all outcomes. In a more practical sense, if we say an event follows a uniform distribution, each day or instance within the defined interval is equally likely to occur. For example, when rolling a fair die, each of the six faces is equally likely to result, representing a uniform distribution. In the given exercise, the assumption made is that births occur uniformly throughout the 1461 days in a four-year cycle. Under this assumption, the chance of a birth on any specific day—like February 29th—is the same across the entire period. While this simplifies calculations, it's vital to acknowledge that real-world data, influenced by various factors, might deviate from this simplistic model.

Birth Rates

Birth rates refer to the number of births occurring within a specific population over a certain period of time. Various elements, such as cultural practices, climate, and even societal norms, can influence these rates, possibly affecting when and how frequently births occur. In examining the leap year exercise, assuming a constant birth rate across all days may not accurately reflect reality. Factors such as planned pregnancies, cultural preferences for certain birth timings, or even natural phenomena like seasonal variations could shift the frequency of births. These elements could impact the probability distribution and cause deviations from an idealized uniform distribution.