Problem 51 - A First Course in Probability

Question

A worker has asked her supervisor for a letter of recommendation for a new job. She estimates that there is an 80 percent chance that she will get the job if she receives a strong recommendation, a 40 percent chance if she receives a moderately good recommendation, and a 10 percent chance if she receives a weak recommendation. She further estimates that the probabilities that the recommendation will be strong, moderate, and weak are .7 \(.2,\) and \(.1,\) respectively. (a) How certain is she that she will receive the new job offer? (b) Given that she does receive the offer, how likely should she feel that she received a strong recommendation? a moderate recommendation? a weak recommendation? (c) Given that she does not receive the job offer, how likely should she feel that she received a strong recommendation? a moderate recommendation? a weak recommendation?

Step-by-Step Solution

Answer

(a) The worker is 65% certain that she will receive the new job offer. (b) Given that she receives the offer, she should feel 86.15% likely that she received a strong recommendation, 12.31% likely that she received a moderate recommendation, and 1.54% likely that she received a weak recommendation. (c) Given that she does not receive the job offer, she should feel 40% likely that she received a strong recommendation, 34.29% likely that she received a moderate recommendation, and 25.71% likely that she received a weak recommendation.

Step 1: Use the law of total probability

The law of total probability states that the probability of an event occurring is the sum of the probabilities of the event occurring given each possible result in the problem. In this case, we want to find the overall probability of the worker receiving a new job offer, P(Job Offer), which is the sum of the probabilities of receiving a job offer given a strong, moderate, or weak recommendation. The formula for the law of total probability is: P(Job Offer) = P(Job Offer | Strong) * P(Strong) + P(Job Offer | Moderate) * P(Moderate) + P(Job Offer | Weak) * P(Weak)

Step 2: Calculate the overall probability

Plug in the given probabilities and calculate the overall probability of receiving a new job offer: P(Job Offer) = (0.8 * 0.7) + (0.4 * 0.2) + (0.1 * 0.1) = 0.56 + 0.08 + 0.01 = 0.65 (a) The probability of the worker receiving a new job offer is 65%. (b) Calculate the probability of receiving a strong, moderate, or weak recommendation given that the worker receives the job offer:

Step 3: Use Bayes' theorem for each type of recommendation

Bayes' theorem can be used to calculate the probability: P(Recommendation | Job Offer) = (P(Job Offer | Recommendation) * P(Recommendation)) / P(Job Offer) For a strong recommendation: P(Strong | Job Offer) = (0.8 * 0.7) / 0.65 ≈ 0.8615 For a moderate recommendation: P(Moderate | Job Offer) = (0.4 * 0.2) / 0.65 ≈0.1231 For a weak recommendation: P(Weak | Job Offer) = (0.1 * 0.1) / 0.65 ≈ 0.0154 (b) Given that the worker receives the job offer, the probability she received a strong recommendation is 86.15%, a moderate recommendation is 12.31%, and a weak recommendation is 1.54%. (c) Calculate the probability of receiving a strong, moderate, or weak recommendation given that the worker does not receive the job offer:

Step 4: Use Bayes' theorem for each type of recommendation and the probability of not receiving the job offer

Since P(No Job Offer) = 1 - P(Job Offer), we can use that to calculate the probabilities of each recommendation type. For a strong recommendation: P(Strong | No Job Offer) = (0.2 * 0.7) / 0.35 ≈ 0.4 For a moderate recommendation: P(Moderate | No Job Offer) = (0.6 * 0.2) / 0.35 ≈ 0.3429 For a weak recommendation: P(Weak | No Job Offer) = (0.9 * 0.1) / 0.35 ≈ 0.2571 (c) Given that the worker does not receive the job offer, the probability she received a strong recommendation is 40%, a moderate recommendation is 34.29%, and a weak recommendation is 25.71%.

Key Concepts

Law of Total Probability

The Law of Total Probability is a foundational rule in probability theory that connects marginal probabilities to conditional probabilities. It states that the total probability of an outcome can be determined by considering all possible ways that outcome can occur. For example, if you want to calculate the probability of an event, such as receiving a job offer, you have to account for all the scenarios that might lead to this event, and then sum their associated probabilities.

Formally, if we have a set of mutually exclusive and exhaustive events, say B1, B2, ..., Bn, and we want to find the probability of an event A, the law of total probability tells us that we can calculate it as:
\[ P(A) = P(A | B1)P(B1) + P(A | B2)P(B2) + \dots + P(A | Bn)P(Bn) \]

In simpler terms, it allows us to break down a complicated problem into simpler parts, which is particularly helpful when direct calculations of probability are difficult. Using this law can vastly simplify probability calculations, as it was the case in the textbook exercise where the likelihood of the job offer was determined.

Bayes' Theorem

Bayes’ theorem is a powerful equation in probability theory that allows us to update our beliefs or probabilities based on new evidence. This theorem helps in finding the reverse probabilities - in other words, calculating the probability of the cause given the outcome. This is incredibly useful in a wide array of fields, from medical diagnosis to machine learning algorithms.

The formal expression of Bayes’ theorem is:
\[ P(A | B) = \frac{P(B | A)P(A)}{P(B)} \]

It says that if we're interested in finding the probability of event A after observing event B, we can do this by understanding how likely B is if A happens, the initial likelihood of A, and how likely is B overall. In the exercise, Bayes’ theorem was used to calculate the posterior probabilities of receiving a strong, moderate, or weak recommendation, given the outcome of receiving or not receiving the job offer.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. The symbol '|' is used to denote 'given that', so the probability of event A occurring, given event B, is written as P(A | B).

This concept is central to many statistical methods and is expressed mathematically as:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \],

provided that P(B) is not zero. This equation signifies that the probability of A happening, on the condition B has occurred, is the proportion of the likelihood of both events happening together to the likelihood of B occurring on its own. In our exercise scenario, conditional probability allowed us to consider the chances of getting a job offer under the condition of receiving a certain type of recommendation.

Probability Calculations

Probability calculations are the bread and butter of making predictions about uncertain events. They involve applying basic probability rules, such as addition and multiplication rules, and more advanced principles like the Law of Total Probability and Bayes' Theorem, to compute the likelihood of various outcomes.

Probability can be framed as a fraction or a percentage, and it ranges from 0 (impossible event) to 1 (a certain event). In context, probability calculations helped us to determine the chance of a worker getting a job offer based on different levels of recommendations. To tackle complex problems, as shown in the exercise, we break them down step by step, calculate individual probabilities, and combine them using the aforementioned concepts to obtain the final probabilities for each scenario.