91Ó°ÊÓ

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 only about 1 in \(2.1\) million births is a leap year baby born to a leap year mom (a probability of 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

Compute the Probability

The reasoning behind the probability calculation would be considering the chances of being born on February 29, which happens only once in 1461 days due to the cycle of leap years. Hence the probability for one person to be born on this day is \(1/1461\). Now, for a person born on such a day to have a child also born on February 29, the probability will be \(1/1461 * 1/1461\), which is roughly \(0.00000047\).

02

Reasoning if the Birth is Equally Likely to Occur any Day

The assumption that a birth is equally likely to occur on any of the 1461 days in a four-year period appears to be broadly reasonable. It presumes that there are no factors that materially sway births to occur on particular days over others, which is a generally consistent assumption, barring any specific cultural or personal beliefs or practices that might target certain days for childbirths.

03

Assessment of the Probability

Based on the reasoning above, it can be inferred that the probability of \(0.00000047\) might be about right. The assumption that birth can occur on any day seems reasonable, and albeit certain external factors can slightly influence this, the assumption and hence the calculated probability seem to be a fair estimation.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Leap Year Statistics

Leap years have long fascinated statisticians and mathematicians due to their unique inclusion of an extra day, February 29th, added nearly every four years to help synchronize the calendar year with the astronomical year. The statistical odds of any event occurring on this day, such as a person being born, are reduced because of its infrequency—happening only once in 1,461 days, since a leap year comes after a cycle of three regular years.

To calculate the probability of a leap day baby becoming a leap year mother, one starts by understanding the individual probability, which is \(1/1461\). However, when considering the independent probability of two such rare events happening in sequence—such as a leap year baby then giving birth on a leap day—the calculations become \(1/1461 * 1/1461\), leading to a much smaller number, significantly illustrating the rarity of the event reported.

Birth Probability

Determining the probability of births on any given day involves understanding the conditions under which babies are born. Assuming that there is an equal chance of birth on each day might overlook factors like planned C-sections, induced labor, natural birth timing tendencies, and cultural factors affecting the timing of births.

This assumption however simplifies the complex variables involved in human birth. Under this assumption, the probability of anyone being born on February 29th is as likely as being born on any other day, making it \(1/1461\), due to the fact that February 29th appears only once in the four-year cycle. When discussing the birth probability of a leap year baby to a leap year mom, the improbability is squared, reflecting the rarity of two successive leap day births.

Statistical Assumptions

In every statistical calculation, assumptions play a crucial role in simplifying real-world complexities into calculable probabilities. The assumption that the spokesperson from the hospital made—that a birth is equally likely to happen on any day within a four-year period—is a simplification necessary to perform the calculation without accounting for the myriad factors that can affect birth timing.

While this may broadly hold true and allow for the calculation of a simplified probability, it is important to consider the potential impact of any deviation from this assumption in real life. Factors like leap year superstitions, seasonal birth rates, or medical interventions could skew the actual probability. In this light, the statistical assumption acts as a baseline, recognizing that in practice, the specific probability may be slightly more or less than the calculated estimate of \(0.00000047\).