Problem 74 - Algebra and Trigonometry

Question

On New Year's Eve, the probability of a person driving while intoxicated or having a driving accident is \(0.35 .\) If the probability of driving while intoxicated is 0.32 and the probability of having a driving accident is \(0.09,\) find the probability of a person having a driving accident while intoxicated.

Step-by-Step Solution

Answer

The probability of a person having a driving accident while intoxicated is 0.06.

Step 1: Identify and Match Probabilities to Elements in the Formula

Based on the information given: probability of a person driving while intoxicated, P(A), is 0.32; the probability of a person having a driving accident, P(B), is 0.09 and the probability of driving while intoxicated or having a driving accident, P(A or B), is 0.35.

Step 2: Plug the Values into the Formula

Apply these values to the formula P(A and B) = P(A) + P(B) – P(A or B), we obtain P(A and B) = 0.32 + 0.09 - 0.35.

Step 3: Perform the Calculations

Calculate the value as follows: P(A and B) = 0.32 + 0.09 - 0.35 = 0.06. This result represents the joint probability of a person driving while intoxicated and having a driving accident.

Key Concepts

Joint Probability

Joint probability refers to the likelihood of two events occurring simultaneously. It is a fundamental concept in probability theory and is particularly useful when we want to understand how two or more events intersect. In this scenario, we are interested in finding the probability of a person both driving while intoxicated and having a driving accident. To determine the joint probability, we use the formula: \[ P(A \text{ and } B) = P(A) + P(B) - P(A \text{ or } B) \]Where:

• \( P(A) \) is the probability of event A (driving while intoxicated).
• \( P(B) \) is the probability of event B (having a driving accident).
• \( P(A \text{ or } B) \) is the probability of either event A or event B occurring.

This formula helps us compile the values of the individual probabilities and subtract the overlap to avoid double-counting.

Probability Calculation

Probability calculation involves systematically using given data to find the likelihood of an event or combination of events. Calculating probability can involve various formulas, depending on the circumstances. In this exercise, we used the values derived from probabilities that are already known:

• \( P(A) = 0.32 \), the likelihood of driving while intoxicated.
• \( P(B) = 0.09 \), the likelihood of having a driving accident.
• \( P(A \text{ or } B) = 0.35 \), the likelihood of either scenario occurring.

By inserting these into the joint probability formula, \[ P(A \text{ and } B) = 0.32 + 0.09 - 0.35 \], we effectively calculated the probability of both events happening at the same time.This method is crucial for avoiding mistakes and ensures accurate results in probability analyses.

Conditional Probability

Conditional probability is about finding the probability of an event occurring given that another event has already happened. Although not directly required in this problem, conditional probability can help us understand related probability interactions. For example, if we knew someone was already intoxicated, we might wish to know the probability of them also having an accident.The concept is formalized by the equation:\[ P(A|B) = \frac{P(A \text{ and } B)}{P(B)} \]Where:

• \( P(A|B) \) is the conditional probability of A given B.
• \( P(A \text{ and } B) \) is the joint probability of events A and B.
• \( P(B) \) is the probability of event B.

Using conditional probability, one can gain insight into how one event impacts the likelihood of another, providing a deeper understanding of probabilistic relationships.