Question

A friend who works in a big city owns two cars, one small and one large. Three-quarters of the time he drives the small car to work, and one-quarter of the time he takes the large car. If he takes the small car, he usually has little trouble parking and so is at work on time with probability \(0.9 .\) If he takes the large car, he is on time to work with probability 0.6 . Given that he was at work on time on a particular morning, what is the probability that he drove the small car?

Short answer

The probability that the friend drove the small car given he was on time is approximately \( 0.818 \).

Step-by-step solution

Identify given probabilities

First, identify the probabilities given in the problem. The prior probabilities are the probabilities of the friend driving the small or large car: \( P(Small) = 0.75 \) and \( P(Large) = 0.25 \). The likelihood probabilities are the probabilities of him being on time given he drove either car: \( P(Time|Small) = 0.9 \) and \( P(Time|Large) = 0.6 \)

Apply the Total Probability theorem

Calculate the total probability of the friend being on time, regardless of the car he drove. This is computed using the Total Probability theorem: \( P(Time) = P(Small)P(Time|Small) + P(Large)P(Time|Large) = 0.75 * 0.9 + 0.25 * 0.6 = 0.825 \)

Use Bayes' theorem

Bayes' theorem allows us to 'reverse' conditional probabilities. It states: \( P(A|B) = P(B|A)P(A) / P(B) \). To find the probability of the friend driving the small car given he was on time \( P(Small|Time) \), we can plug into the Bayes' theorem: \( P(Small|Time) = P(Time|Small)P(Small) / P(Time) = (0.9 * 0.75) / 0.825 = 0.818 \)

Key concepts

Conditional Probability

Understanding conditional probability is crucial when dealing with events that are affected by other events. It refers to the probability of an event occurring, given that another event has already occurred. For instance, if you want to know the likelihood that a friend is driving their small car given they arrive at work on time, you're considering conditional probability.

In mathematical terms, the conditional probability of Event A given Event B has occurred is denoted by \( P(A|B) \). This can be computed by dividing the probability of both events happening, \( P(A \cap B) \), by the probability of Event B: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \].

Applied to our example, we look for \( P(Small|Time) \), which requires us to know both \( P(Small) \) and \( P(Time|Small) \), along with \( P(Time) \), the probability of being on time irrespective of the car chosen. By understanding conditional probability, we can quantify the influence of one event on the other.

Total Probability Theorem

The Total Probability theorem is a backbone of probability theory that allows us to calculate the likelihood of a particular outcome based on several different possible events that could lead to this outcome.

It essentially breaks down a complex event into simpler, mutually exclusive events. The theorem expresses the probability of a target event, \( P(B) \), as the sum of the probabilities of it occurring via various independent paths, or causes: \[ P(B) = \sum_{i}P(A_i)P(B|A_i) \], where \( P(A_i) \) are the prior probabilities of each cause and \( P(B|A_i) \) are the conditional probabilities.

For the car example, we use this theorem to calculate \( P(Time) \), by considering all possible ways the friend could be on time — either by driving the small or the large car. The Total Probability theorem simplifies the complex real-world scenarios into manageable calculations.

Prior Probability

Prior probability, often just called the 'prior,' is the raw probability of an event before any additional information is considered. It is not influenced by any new data and is often based on historical information, theoretical assumptions, or inherent frequencies.

Using our example, the prior probabilities are simply the likelihoods of the friend choosing a small or large car without any information about whether he was on time or not: \( P(Small) = 0.75 \) and \( P(Large) = 0.25 \).

Priors are essential building blocks when computing updated probabilities using Bayes' theorem because they represent our initial beliefs about an event's likelihood before we factor in the specifics of a given situation. They are the starting point of analyzing many probability problems.

Likelihood Probability

Likelihood probability is akin to conditional probability, yet it often pertains to parameters of a statistical model and represents how likely specific observed events are, given particular parameter values. In Bayes' theorem, the likelihood is the evidence that can change our view on the prior probability.

In our exercise, the likelihood probabilities are the chances of arriving on time given the choice of car: \( P(Time|Small) = 0.9 \) and \( P(Time|Large) = 0.6 \). That tells us how 'fitting' or 'compatible' the prior choices are with the evidence of being on time. Likelihood serves as an update factor that adjusts our initial beliefs (prior probabilities) in light of new evidence.

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random phenomena. It provides the foundations for reasoning about uncertainty and quantifying how likely events are to occur. Central concepts include random variables, events, and the probabilities associated with them.

Through the lens of probability theory, we tackle real-world randomness from dice rolls to stock market fluctuations. Our friend's car choice scenario represents a practical application of this theory: the randomness of daily events like traffic and parking complexity are distilled into calculable probabilities. Understanding the basics of probability theory enables students to dissect complex problems, make informed decisions amidst uncertainty, and appreciate the patterns in seemingly random events.