Problem 33 - A First Course in Probability

Question

On rainy days, Joe is late to work with probability \(.3 ;\) on nonrainy days, he is late with probability.1. With probability.7, it will rain tomorrow. (a) Find the probability that Joe is early tomorrow. (b) Given that Joe was early, what is the conditional probability that it rained?

Step-by-Step Solution

Answer

(a) The probability that Joe is early tomorrow is \(P(L') = 0.6\) (b) Given that Joe was early, the conditional probability that it rained is \(P(R | L') ≈ 0.667\)

Step 1: Find P(L') using the Law of Total Probability

Using the Law of Total Probability, we can find P(L') as follows: P(L') = P(L' and R) + P(L' and R') As we know P(L) and P(R) we can find P(L' and R') and P(L' and R) using complementary probabilities: P(L' and R) = P(R) - P(R and L) = 0.7 - 0.3 = 0.4 P(L' and R') = P(R') - P(R' and L) = (1 - 0.7) - 0.1 = 0.2 Now we can find P(L'): P(L') = P(L' and R) + P(L' and R') = 0.4 + 0.2 = 0.6

Step 2: Find P(R | L') using Bayes' theorem

Bayes' theorem states that P(R | L') = P(L' | R) * P(R) / P(L') First, we need to find the conditional probability P(L' | R). P(L' | R) = P(L' and R) / P(R) = 0.4 / 0.7 ≈ 0.571 Now, we can find P(R | L') using the Bayes' theorem: P(R | L') = P(L' | R) * P(R) / P(L') ≈ 0.571 * 0.7 / 0.6 ≈ 0.667

Step 3: Answer

(a) The probability that Joe is early tomorrow is P(L') = 0.6 (b) Given that Joe was early, the conditional probability that it rained is P(R | L') ≈ 0.667

Key Concepts

Law of Total Probability

The Law of Total Probability is a fundamental rule in Probability Theory that enables us to find the probability of a particular event by breaking it down into several mutually exclusive events. Just like in the exercise where Joe can be either late or early (but not both), this law is particularly useful in complex scenarios where different outcomes may occur under various conditions.

Let's say we want to find the probability that Joe is early to work. According to the Law of Total Probability, we should consider all the ways in which Joe can be early. In this case, there are two mutually exclusive scenarios: it can either rain or not rain. We find the probability of Joe being early for both scenarios and add them together to get the total probability of him being early.

This approach simplifies complex problems by breaking them down into simpler components that are easier to handle, a technique highly recommended for grasping the probability of compound events.

Bayes' theorem

Bayes' theorem is a powerful formula used to calculate conditional probabilities. It allows us to update our beliefs about the likelihood of an event based on new evidence. The theorem essentially tells us how likely an event is, given that another related event has occurred.

For instance, in our exercise about Joe, we use Bayes' theorem to determine the probability it rained given that Joe was early. Essentially, this formula lets us flip our conditional probabilities. We may know the probability of Joe being early given that it rained, but Bayes' theorem allows us to infer the reverse – calculating the probability that it rained given Joe's earliness.

Quick Tip

Bayes' theorem is particularly useful when dealing with diagnostic tests, such as medical screenings or even figuring out the likelihood of certain events based on past occurrences, making it a go-to tool in statistics and data science.

Probability Theory

Probability Theory is the branch of mathematics focused on analyzing random events and determining the likelihood of different outcomes. It is based on the idea that certain events occur with definable frequencies over large numbers of trials. In the context of our example with Joe, Probability Theory lays the groundwork to predict the likelihood of Joe being late or early under varying weather conditions.

In learning Probability Theory, it's crucial to grasp various concepts such as independent and dependent events, mutually exclusive outcomes, and the range of probabilities from 0 (impossible event) to 1 (certain event). Understanding these concepts allows students to interpret and calculate probabilities for complex scenarios in their everyday lives and academic studies.

By practicing exercises like the one we've discussed, where multiple factors influence the outcome, learners can deepen their understanding of how to use Probability Theory effectively.

Complementary probabilities

Complementary probabilities refer to the rule that the probability of an event occurring plus the probability of the event not occurring always equals 1. This is because something either happens or it doesn't – there are no other options. In the case of Joe being late or early, if we know the probability of him being late, we can easily find the probability of the opposite situation, being early, by subtracting from 1.

In our exercise example, we use the complementary probability to deduce that if Joe has a 30% chance of being late on rainy days, there's a 70% chance of him being early. Complementary probabilities are useful for checking the accuracy of your calculations and they also provide a quick way to find missing probability values without extensive computation.