"[DATA] A Random Process: Green Lights On your drives to school each day you feel like there is a light that is always red when you reach it. You decide to record data to determine the likelihood of arriving at the light while it is red. Open the data set 5_1_38 at www.pearsonhighered.com/sullivanstats, which contains the day number and whether the light was red (1), or not (0), for 120 consecutive days in which you drove to school.
f. What is the estimate of the probability of the light being red when you reach the intersection?"

Verified step by step guidance

1. Understand that the problem asks for the estimated probability that the traffic light is red when you arrive at the intersection. This is a relative frequency probability based on observed data.
2. Identify the total number of days recorded, which is given as 120 days.
3. Count the number of days the light was red. In the data set, red is indicated by a 1, so sum all the 1's to find the total number of red light occurrences.
4. Calculate the estimated probability by dividing the number of days the light was red by the total number of days. Mathematically, this is expressed as: \[\text{Estimated Probability} = \frac{\text{Number of days light was red}}{\text{Total number of days}}\]
5. Interpret the result as the likelihood or chance that the light will be red when you reach the intersection on any given day, based on your recorded data.

Key Concepts

Probability Estimation

Probability estimation involves calculating the likelihood of an event based on observed data. In this case, it is estimated by dividing the number of days the light was red by the total number of days observed. This relative frequency approach provides an empirical probability reflecting real-world outcomes.

Random Process

A random process is a sequence of random variables representing outcomes over time or space. Here, the daily observation of the traffic light being red or not is modeled as a random process, where each day's outcome is a random event influenced by various factors.

Binary Data and Indicator Variables

Binary data consists of two possible outcomes, often coded as 0 or 1. Indicator variables are used to represent such data, with 1 indicating the event of interest (light is red) and 0 otherwise. This simplifies analysis and probability calculations for events with two outcomes.