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Chapter 12: Problem 12

Cars leaving the highway can take a right turn (R), left turn \((L),\) or go straight \((S)\) You are collecting data on traffic patterns at this intersection and you group your observations by taking four cars at a time every 5 minutes. a) List a few outcomes in your sample space U. How many are there? b) List the outcomes in the event that all cars go in the same direction. c) List the outcomes that only two cars turn right. d) List the outcomes that only two cars go in the same direction.

Short Answer

Expert verified
(a) 81 outcomes. (b) (R,R,R,R), (L,L,L,L), (S,S,S,S). (c) Multiple outcomes such as (R,R,L,L). (d) Examples include (R,R,L,S), (L,L,R,S), etc.

Step by step solution

01

Understanding the Scenario

We are observing a traffic intersection where cars can take three possible directions: Right (R), Left (L), or Straight (S). We take observations of four cars at a time every 5 minutes to analyze the traffic patterns.
02

Listing Sample Space Outcomes

Each car has 3 choices (R, L, S), and since we are observing four cars, the total outcomes in the sample space can be calculated as \(3^4\). This results in 81 possible outcomes. Examples of outcomes are: (R, R, R, R), (L, L, L, L), (S, S, S, S), and (R, L, S, R).
03

Event of All Cars in the Same Direction

The event that all cars go in the same direction includes only three outcomes: (R, R, R, R), (L, L, L, L), and (S, S, S, S).
04

Event of Only Two Cars Turning Right

For the event where exactly two out of the four cars turn right (R), we fix two positions as R and select the remaining two as either L or S:- Combinations such as (R, R, L, L), (R, R, S, L), (R, L, R, S), etc.- Calculate using combination and permutation means resulting in (\(\binom{4}{2} imes \binom{2}{2}\) = 6 pairs of R, with two other possibilities each).
05

Event of Only Two Cars Going in the Same Direction

To have two cars in the same direction, you fix two cars, and choose two other positions randomly. Examples include: - Two R's with the remaining being L or S, like (R, R, L, S). Each remaining can have the opposite directions like L or S. The same goes for L and S with unique distinct selections as needed.

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Key Concepts

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

Probability Outcomes
In probability and statistics, outcomes refer to all the possible results that could occur from a given random phenomenon. In our scenario, we are analyzing an intersection where cars can take three directions (Right, Left, Straight) as they leave the highway. The variety of ways these cars can move represents different outcomes.
  • Each car at the intersection has three possible directions: Right (R), Left (L), and Straight (S).
  • When observing four cars, each independent choice multiplies the total possibilities.
Every combination of these individual decisions represents a distinct outcome in our study of traffic patterns. As such, the analysis of such outcomes informs decision-making and understanding of traffic dynamics.
Combinatorics
Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of elements within a set. This concept is crucial in understanding our traffic pattern exercise.Considering four cars, each with three choices, combinatorics helps us determine the total number of possible combinations of outcomes.
  • The formula used here is based on exponentiation: each car鈥檚 three choices raised to the power of the four cars, resulting in \(3^4\).
  • This calculation gives us a total of 81 different combinations of traffic patterns.
Combinatorics also helps calculate the number of specific event outcomes. For example, if we want to know how many ways exactly two cars can turn right, combinatorics provides the necessary tools to calculate this and consider all potential arrangements.
Sample Space
The sample space is a comprehensive set of all possible outcomes of a random experiment. In the context of traffic patterns, the sample space encompasses every possible way the four cars might either turn right, turn left, or go straight. To find the sample space size in this scenario, we employ:
  • Calculating the formula \(3^4 = 81\), providing us with all possible arrangements.
  • Each unique combination like (R, R, L, S) is a point in this sample space, representing a specific traffic pattern observed during the data collection.
By understanding the complete sample space, analysts can better assess and predict likely traffic outcomes, helping to improve planning and traffic management.
Event Probability
Event probability refers to the likelihood of a specific event happening within a sample space. In this case, an event is a particular arrangement or outcome of the four cars at the intersection. For instance, when calculating the event where all cars go in the same direction, we see that:
  • There are only three outcomes: (R, R, R, R), (L, L, L, L), and (S, S, S, S).
  • We calculate the probability by dividing the number of favorable outcomes (3) by the total number of possible outcomes (81).
Understanding event probability allows traffic analysts to make informed guesses about how traffic might behave under different conditions, aiding in effective intersection planning and management strategies.

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