Question

In a certain population of the European starling, there are 5,000 nests with young. The distribution of brood size (number of young in a nest) is given in the accompanying table. \(^{12}\) $$ \begin{array}{|cc|} \hline \text { Brood Size } & \text { Frequency (No. of Broods) } \\ \hline 1 & 90 \\ 2 & 230 \\ 3 & 610 \\ 4 & 1,400 \\ 5 & 1,760 \\ 6 & 750 \\ 7 & 130 \\ 8 & 26 \\ 9 & 3 \\ 10 & 1 \\ \hline \text { Total } & 5,000 \\ \hline \end{array} $$ Suppose one of the 5,000 broods is to be chosen at random, and let \(Y\) be the size of the chosen brood. Find (a) \(\operatorname{Pr}\\{Y=3\\}\) (b) \(\operatorname{Pr}\\{Y \geq 7\\}\) (c) \(\operatorname{Pr}\\{4 \leq Y \leq 6\\}\)

Short answer

Pr{Y=3} = \frac{610}{5,000}, Pr{Y \(geq 7} = \frac{130 + 26 + 3 + 1}{5,000}, Pr{4 \leq Y \leq 6} = \frac{1,400 + 1,760 + 750}{5,000}

Step-by-step solution

Calculate Pr{Y=3}

To find the probability that the randomly chosen brood has exactly 3 young, divide the frequency of broods with 3 young by the total number of broods.

Calculate Pr{Y>=7}

To calculate the probability that the randomly chosen brood has 7 or more young, sum the frequencies of broods having 7, 8, 9, and 10 young. Then, divide this sum by the total number of broods.

Calculate Pr{4

To find the probability that the randomly chosen brood has between 4 and 6 young (inclusive), add the frequencies for broods with 4, 5, and 6 young. Divide this sum by the total number of broods.

Key concepts

Brood Size Frequency

The term 'brood size frequency' refers to how often different brood sizes occur within a given population. In our context, it represents the number of European starling nests with a specific number of young. Understanding the frequency data is crucial as it forms the basis for analyzing population trends and making predictions about future distributions. When examining such data, it's essential to look not only at the sheer numbers but also at the relative frequencies, which provide insight into the most common or rare sizes of broods within the population.

• The table shows that smaller broods are quite common, with 90 occurrences of a single chick in a nest.
• As the brood size increases, the frequency generally decreases, which is typical of many biological populations.
• Large broods, like those with 10 young, are exceptionally rare, with only one occurrence in the population.

The frequency data serves as a starting point for calculating probabilities of various brood sizes when a nest is chosen at random, demonstrating a real-world application of probability theory in biology and ecology.

Random Sampling

Random sampling is a fundamental concept used in statistics to draw conclusions about a population. It is the process of selecting a subset of individuals, items, or data points from a larger set, where each member has an equal chance of being chosen. In our exercise, the random sample is one brood from the 5,000 available. This randomness ensures that the probability calculations are unbiased and representative of the population, providing a fair estimation of what can be expected on average.

• In practice, random sampling can be achieved by various methods, including random number generators or drawing lots.
• It is particularly useful in ecological studies where examining the entire population may be impossible or impractical.
• Random sampling underlies the principles of inferential statistics, where we make estimations or predictions about a population based on a sample.

In the case of European starlings, using random sampling helps us estimate the likelihood of encountering a nest with a particular number of chicks, which could then be used for developing conservation strategies or understanding population dynamics.

Probability Calculation

Probability calculation is used to determine the likelihood of a specific event occurring. This is a key concept within the field of statistics and is essential for interpreting data distributions such as our brood size frequencies. The steps to calculate probability generally involve dividing the number of favorable outcomes (the specific event we're interested in) by the total number of possible outcomes.

• For example, to calculate the probability of selecting a nest with exactly three young (Pr{Y=3}), we divide the number of broods with three young by the total number of broods.
• For cumulative probabilities, such as finding the chance of a brood being larger or equal to seven (Pr{Y>=7}), we sum the relevant frequencies before division.
• To find the probability of a brood size falling within a range (Pr{4 <= Y <= 6}), we total the frequencies for all sizes within the range and then divide by the total number of broods.

This approach can be applied to various fields where uncertainty and randomness play a role. By understanding probability calculations, students can better grasp not only exercises like the one provided but also real-life situations where outcomes are uncertain.