91Ó°ÊÓ

The article "Three Sisters Give Birth on the Same Day" (Chance, Spring 2001, 23-25) used the fact that three Utah sisters had all given birth on March 11, 1998 as a basis for posing some interesting questions regarding birth coincidences. a. Disregarding leap year and assuming that the other 365 days are equally likely, what is the probability that three randomly selected births all occur on March 11? Be sure to indicate what, if any, extra assumptions you are making. b. With the assumptions used in part (a), what is the probability that three randomly selected births all occur on the same day? c. The author suggested that, based on extensive data, the length of gestation (time between conception and birth) could be modeled as having a normal distribution with mean value 280 days and standard deviation \(19.88\) days. The due dates for the three Utah sisters were March 15, April 1, and April 4, respectively. Assuming that all three due dates are at the mean of the distribution, what is the probability that all births occurred on March 11? d. Explain how you would use the information in part (c) to calculate the probability of a common birth date.

Step by step solution

01

Calculate the Probability for Part (a)

To calculate the probability that three randomly selected births all occur on March 11, we assume the probability of birth on any given day (not considering leap years) is \( \frac{1}{365} \). Thus, the probability that all three births occur on the same specified day (March 11) is given by multiplying this probability three times:\[ \left( \frac{1}{365} \right)^3 = \frac{1}{365 \times 365 \times 365} = \frac{1}{365^3} \approx 2.78 \times 10^{-8}. \] We assume independence of birth dates and a uniform distribution across the year.

02

Calculate the Probability for Part (b)

The probability that all three randomly selected births occur on the same (but unspecified) day requires accounting for any of the 365 days. Each day has the probability \( \frac{1}{365^3} \), and since there are 365 possible days, the probability is:\[ 365 \times \frac{1}{365^3} = \frac{1}{365^2} = \frac{1}{133225} \approx 7.51 \times 10^{-6}. \]

03

Set Up the Problem for Part (c)

Using the normal distribution to model gestation times with a mean of 280 days and a standard deviation \( \sigma = 19.88 \) days, we consider each sister's due date. Assuming their due dates are at the mean, we compute how many standard deviations the birth date (March 11) is from the due date for each sister. Due date differences from March 11 to March 15, April 1, and April 4 in days are \(-4\), \(-21\), and \(-24\).

04

Calculate Probability Using Normal Distribution for Part (c)

Convert days to standard deviations by dividing each by the standard deviation. For example, for the first sister:\[ Z_1 = \frac{-4}{19.88} \approx -0.201. \] Translate each Z-value into a probability using the standard normal distribution table. Repeat for two additional Z-values. The probability of less than each Z-value on the standard normal curve is derived from reference tables or software.

05

Probability of All Events for Part (c)

Assuming independence, calculate the product of probabilities that each sister gives birth on March 11. Let \( P(Z < Z_1) \), \( P(Z < Z_2) \), and \( P(Z < Z_3) \) be the probabilities found. Multiply them together:\[ P = P(Z < Z_1) \times P(Z < Z_2) \times P(Z < Z_3). \]

06

Explanation for Part (d)

To calculate the probability that three specific individuals share a birth date, given their due dates fall on three different dates, use the normal distribution as demonstrated. Compute the joint probability that birth deviations align on the same date, aggregating the independent probabilities derived from the Z-scores as done previously.

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.

Coincidence

Coincidence often refers to events that seem related but are actually independent of one another. In probability, it addresses events that occur by chance simultaneously or in a specific relationship. For example, when three sisters give birth on the same day, it may initially seem extraordinary. However, these are independent events that can be explained by probability. Statistical methods help us quantify how likely it is for such an event to happen. By working through probability calculations, we can determine if the event is just a remarkable coincidence or something more.

Uniform Distribution

A uniform distribution indicates that all outcomes are equally likely. In the context of birth dates, if births are assumed to occur randomly on any day of the year, each day has the same probability of being a birth date.

• The year comprises 365 days (not considering leap years).
• The probability that one specific birth occurs on a specific day is therefore \( \frac{1}{365} \).
• When considering multiple births, uniform distribution assumes each birth event is independent from each other.

This assumption allows us to apply simple probability rules to calculate the likelihood of simultaneous events, such as three births occurring on any specific same day.

Normal Distribution

Normal distribution, or the bell curve, is one of the most important concepts in statistics. It's used to model real-valued random variables with known mean and standard deviation. In terms of gestation periods, this curve provides insight into natural variability, allowing predictions about birth timings.

• The mean gestation period is 280 days.
• A standard deviation illustrates typical deviation from the mean (for gestation, \(19.88\) days).
• By converting real dates to Z-scores, we can find probabilities and make predictions.

Normal distribution helps to understand and calculate the probability of a birth occurring on a specific date, using natural variation data.

Gestation Period

The gestation period is the time between conception and birth. For humans, it is usually around 280 days, depending on individual differences. Probabilistic modeling of gestation can provide deeper insights into birth timing variations. The gestation period can follow a normal distribution:

• Mean gestation period: 280 days.
• Standard deviation: \(19.88\) days, explaining typical variability.

By understanding the distribution of gestation periods, we can estimate the probability of births aligning on a specific date. This understanding aids in analyzing unusual birth coincidences.

Z-score

A Z-score, also known as a standard score, indicates how many standard deviations an element is from the mean. It is calculated by dividing the deviation (difference from mean) by the standard deviation: \[ Z = \frac{X - \mu}{\sigma} \]

• \(X\) is the observed value (e.g., a birth date).
• \(\mu\) is the mean.
• \(\sigma\) is the standard deviation.

For the sisters' due dates, a Z-score measures each birth date's distance from the mean gestation period. This information can be used to find probabilities from the normal distribution tables, allowing us to understand the likelihood of a common birth date as a coincidence.