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 job offer, she should feel as follows about the recommendation received: - Strong: 86.2% - Moderate: 12.3% - Weak: 1.5% (c) Given that she does not receive the job offer, she should feel as follows about the recommendation received: - Strong: 40% - Moderate: 34.29% - Weak: 25.71%

Step 1: Identify the given probabilities

We are given the following probabilities: 1. P(gets job | strong recommendation) = 0.8 2. P(gets job | moderate recommendation) = 0.4 3. P(gets job | weak recommendation) = 0.1 4. P(strong recommendation) = 0.7 5. P(moderate recommendation) = 0.2 6. P(weak recommendation) = 0.1

Step 2: Calculate the probability of getting the job

We need to find P(gets job), which we can find using the total probability theorem: P(gets job) = P(gets job | strong recommendation) * P(strong recommendation) + P(gets job | moderate recommendation) * P(moderate recommendation) + P(gets job | weak recommendation) * P(weak recommendation) P(gets job) = (0.8 * 0.7) + (0.4 * 0.2) + (0.1 * 0.1) P(gets job) = 0.56 + 0.08 + 0.01 P(gets job) = 0.65 (a) The worker is 65% certain that she will receive the new job offer.

Step 3: Calculate the probabilities of each type of recommendation given she gets the job

Now we need to find P(strong recommendation | gets job), P(moderate recommendation | gets job), and P(weak recommendation | gets job). We do this using Bayes' theorem: P(strong recommendation | gets job) = P(gets job | strong recommendation) * P(strong recommendation) / P(gets job) P(strong recommendation | gets job) = (0.8 * 0.7) / 0.65 P(strong recommendation | gets job) = 0.862 P(moderate recommendation | gets job) = P(gets job | moderate recommendation) * P(moderate recommendation) / P(gets job) P(moderate recommendation | gets job) = (0.4 * 0.2) / 0.65 P(moderate recommendation | gets job) = 0.123 P(weak recommendation | gets job) = P(gets job | weak recommendation) * P(weak recommendation) / P(gets job) P(weak recommendation | gets job) = (0.1 * 0.1) / 0.65 P(weak recommendation | gets job) = 0.015 (b) Given that she receives the job offer, she should feel as follows about the recommendation received: - Strong: 86.2% - Moderate: 12.3% - Weak: 1.5%

Step 4: Calculate the probabilities of each type of recommendation given she does not get the job

Next, we need to find the probabilities of getting a strong, moderate, or weak recommendation given that she does not get the job. Since we know P(gets job), we can find P(does not get job) by subtracting P(gets job) from 1: P(does not get job) = 1 - P(gets job) P(does not get job) = 1 - 0.65 P(does not get job) = 0.35 Now we calculate the conditional probabilities: P(strong recommendation | does not get job) = P(does not get job | strong recommendation) * P(strong recommendation) / P(does not get job) P(strong recommendation | does not get job) = (1 - 0.8) * 0.7 / 0.35 P(strong recommendation | does not get job) = 0.6 / 0.35 P(strong recommendation | does not get job) = 0.4 P(moderate recommendation | does not get job) = P(does not get job | moderate recommendation) * P(moderate recommendation) / P(does not get job) P(moderate recommendation | does not get job) = (1 - 0.4) * 0.2 / 0.35 P(moderate recommendation | does not get job) = 0.12 / 0.35 P(moderate recommendation | does not get job) = 0.3429 P(weak recommendation | does not get job) = P(does not get job | weak recommendation) * P(weak recommendation) / P(does not get job) P(weak recommendation | does not get job) = (1 - 0.1) * 0.1 / 0.35 P(weak recommendation | does not get job) = 0.09 / 0.35 P(weak recommendation | does not get job) = 0.2571 (c) Given that she does not receive the job offer, she should feel as follows about the recommendation received: - Strong: 40% - Moderate: 34.29% - Weak: 25.71%

Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory. Imagine you are trying to predict the outcome of an event, but you have some additional information that might influence the outcome. This is where conditional probability comes into play. It helps you calculate the probability of an event occurring, given that another event has already occurred.

In the context of our exercise, let’s think about the probability of landing a new job given the type of recommendation letter received. For example:

• The probability of securing the job given a strong recommendation is expressed as \( P(\text{gets job | strong recommendation}) = 0.8 \).
• This means that if you have a strong recommendation, there's an 80% chance of getting the job.

To calculate such probabilities for other types of recommendations, we use the conditional probability formula:\[P(A|B) = \frac{P(A \cap B)}{P(B)}\]Here, \( A \) is the event of getting the job, and \( B \) is the type of recommendation received. Conditional probability allows us to make informed predictions by factoring in these additional bits of information.

Total Probability Theorem

The Total Probability Theorem is a powerful tool used to calculate the probability of an event based on all possible scenarios that could lead to that event.

In our exercise, we are interested in finding the overall probability of getting the job, denoted by \( P(\text{gets job}) \). To find this probability, we utilize the possible types of recommendations as scenarios. Each recommendation type has its own probability of leading to a job offer:

• Probability of getting the job with a strong recommendation: \( P(\text{gets job | strong recommendation}) = 0.8 \)
• Probability of getting the job with a moderate recommendation: \( P(\text{gets job | moderate recommendation}) = 0.4 \)
• Probability of getting the job with a weak recommendation: \( P(\text{gets job | weak recommendation}) = 0.1 \)

The Total Probability Theorem is applied as follows:\[P(\text{gets job}) = P(\text{gets job | strong recommendation}) \times P(\text{strong recommendation}) \+ P(\text{gets job | moderate recommendation}) \times P(\text{moderate recommendation}) \+ P(\text{gets job | weak recommendation}) \times P(\text{weak recommendation})\]Through this method, we found out that there is a 65% chance that the worker will receive the job. This is a comprehensive way to consider all possible influences on the outcome.

Probability Calculations

Once you have the foundational knowledge of conditional probability and the Total Probability Theorem, the next step involves performing specific probability calculations to solve problems.

In this exercise, we calculated specific probabilities like \( P(\text{strong recommendation | gets job}) \), which uses Bayes' theorem. This theorem helps reverse the conditional probability – meaning, if you know \( P(\text{gets job | strong recommendation}) \), you can find \( P(\text{strong recommendation | gets job}) \).

Bayes' theorem can be expressed as:\[P(B|A) = \frac{P(A|B) \times P(B)}{P(A)}\]For example, knowing:

• \( P(\text{gets job | strong recommendation}) = 0.8 \)
• \( P(\text{strong recommendation}) = 0.7 \)
• \( P(\text{gets job}) = 0.65 \)

You can find that the probability of having received a strong recommendation, given that she got the job, is approximately 86.2%.

Such probability calculations are critical for analyzing outcomes under uncertainty and help in making more informed decisions.