Question

Consider a population of the fruitfly Drosophila melanogaster in which \(30 \%\) of the individuals are black because of a mutation, while \(70 \%\) of the individuals have the normal gray body color. Suppose three flies are chosen at random from the population; let \(Y\) denote the number of black flies out of the three. Then the probability distribution for \(Y\) is given by the following table: $$ \begin{array}{|cc|} \hline Y \text { (No. Black) } & \text { Probability } \\ \hline 0 & 0.343 \\ 1 & 0.441 \\ 2 & 0.189 \\ 3 & 0.027 \\ \hline \text { Total } & 1.000 \\ \hline \end{array} $$ (a) Find \(\operatorname{Pr}\\{Y \geq 2\\}\) (b) Find \(\operatorname{Pr}\\{Y \leq 2\\}\)

Short answer

\( \operatorname{Pr}\{Y \geq 2\} = 0.216 \), \( \operatorname{Pr}\{Y \leq 2\} = 0.973 \).

Step-by-step solution

- Understand the Task

In this exercise, we need to find the cumulative probabilities: (a) The probability that at least 2 flies out of the 3 chosen are black, denoted as \( \operatorname{Pr}\{Y \geq 2\} \), and (b) the probability that at most 2 flies out of the 3 chosen are black, denoted as \( \operatorname{Pr}\{Y \leq 2\} \).

- Calculate \( \operatorname{Pr}\{Y \geq 2\} \)

To find the probability that at least 2 flies are black, we add the probabilities that exactly 2 and exactly 3 flies are black. This can be obtained from the table: \( \operatorname{Pr}\{Y \geq 2\} = \operatorname{Pr}\{Y = 2\} + \operatorname{Pr}\{Y = 3\} = 0.189 + 0.027 = 0.216 \).

- Calculate \( \operatorname{Pr}\{Y \leq 2\} \)

The probability that at most 2 flies are black can be derived by adding the probabilities that 0, 1, or 2 flies are black. We can also arrive at this by subtracting \( \operatorname{Pr}\{Y = 3\} \) from the total probability, which is always 1. This gives us: \( \operatorname{Pr}\{Y \leq 2\} = 1 - \operatorname{Pr}\{Y = 3\} = 1 - 0.027 = 0.973 \). Alternatively, one could add \( \operatorname{Pr}\{Y = 0\} \), \( \operatorname{Pr}\{Y = 1\} \), and \( \operatorname{Pr}\{Y = 2\} \) directly from the table if provided.

Key concepts

Cumulative Probability

Cumulative probability is a fundamental concept in statistics, involving the probability that a random variable takes on a value less than or equal to a particular number. It effectively sums up the probabilities of a set of outcomes.

In the context of the given exercise, where we are dealing with the number of black Drosophila melanogaster fruitflies, the cumulative probability allows us to answer questions about the likelihood of finding a certain number of these flies out of a sample. For instance, to calculate \( \operatorname{Pr}\{Y \leq 2\} \) (the probability of finding at most two black flies), we sum the probabilities of finding 0, 1, or 2 black flies.

Understanding cumulative probabilities is crucial not only for solving this type of problem but also for real-world applications such as determining distribution tails in quality control, finance, or meteorological forecasts.

Binomial Distribution

The binomial distribution is a specific probability distribution that models the number of successes in a fixed number of independent trials, with the same probability of success in each trial.

For example, each fruitfly in the given problem can be considered a trial. Black or not black is a success or failure, respectively, and we're interested in the number of successes (black flies) out of three trials (flies). The requirements of binomiality are met since each selection is independent, and there are only two possible outcomes.

Understanding binomial distribution helps in predicting outcomes in scenarios like flipping a coin, quality testing in manufacturing, and countless others where the 'success-failure' model applies.

Probability Theory

Probability theory is the branch of mathematics concerned with the analysis of random phenomena. It provides the theoretical underpinning for statistics, which is used in this exercise to determine the likelihood of occurrence of different numbers of black flies.

Critical for interpreting and understanding data, probability theory encompasses concepts such as random variables, distributions like the binomial, and cumulative probabilities. These concepts are interconnected, and knowing how to use them in combination allows us to make informed predictions and decisions based on incomplete information or in the presence of randomness.