Unit: Independent and Conditional Probability

The Lesson Activities will help you meet these educational goals:

Mathematical PracticesUse mathematics to model real-world situations, use appropriate tools strategically, and look for and make use of structure.

InquiryYou will perform an investigation in which you will make observations, analyze results, and communicate your results in written form.

STEMYou will use mathematical processes and analysis in scientific investigation, analyze real-world situations, and gain insight into careers in science, technology, engineering, and math.

21st Century SkillsYou will apply creativity and innovation, use critical-thinking and problem-solving skills, communicate effectively, assess and validate information, perform large-scale data analysis, and carry out technology-assisted modeling.

Introduction

Suppose you're an actuary working for a life insurance company and are evaluating eight potential clients for life insurance policies:

Jacob is a newborn non-Hispanic white male (30-year policy)

Carol is a 44-year-old non-Hispanic black female (20-year policy)

Geraldo is a 25-year-old Hispanic male (30-year policy)

Meg is a 39-year-old Asian female (20-year policy)

Earvin is a 68-year-old non-Hispanic black male (10-year policy)

Dora is a 53-year-old Hispanic female (20-year policy)

Adam is an 18-year-old Asian male (30-year policy)

Sally is an 80-year-old non-Hispanic white female (10-year policy)

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Directions and Analysis

Task 1: Finding the Likelihood of Survival of Different Clients

In this task, you will refer to the life tables published by the US government listed at the bottom of page and find the probabilities of your eight potential clients surviving to the end of their policy periods. Make sure you use the appropriate life table for each individual.

a.What is the probability that Jacob, the newborn you're considering for a 30-year policy, lives to be 30 years old? Jacob is a non-Hispanic white male, so look in table 14.

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b.What is the probability that Carol, the 44-year-old you're considering for a 20-year policy, lives to be 64 years old? Note that this is a conditional probability. Another way of saying this is, "What is the probability that Carol turns 64 years old given that she turns 44 years old?" Carol is a non-Hispanic black female, so look in table 18.

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c.What is the probability that Geraldo, the 25-year-old you're considering for a 30-year policy, lives to be 55 years old? Remember that Geraldo is a Hispanic male.

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d.What is the probability that Meg, the 39-year-old you're considering for a 20-year policy, lives to be 59 years old? Meg is an Asian female, but there is no specific life table for Asian females; look in table 3, which is a general table for females.

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e.What is the probability the Earvin, the 68-year-old you're considering for a 10-year policy, lives to be 78 years old? Remember that Earvin is a non-Hispanic black male.

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f.What is the probability that Dora, the 53-year-old you're considering for a 20-year policy, lives to be 73 years old? Remember that Dora is a Hispanic female.

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g.What is the probability that Adam, the 18-year-old you're considering for a 30-year policy, lives to be 48 years old? Adam is an Asian male. There is no specific data for Asian males; look at data in table 2, which is a general table for all males.

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h.What is the probability that Sally, the 80-year old you're considering for a 10-year policy, lives to be 90 years old? Sally is a non-Hispanic white female, so look at the data in table 15.

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i.What is the probability that all of the potential clients would be alive when their respective policies end? Note that each potential client surviving to the end of his or her respective policy is an independent event.

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Task 2: Simulating Client Lives

For this task, use the worksheet labeled Task 2a in this spreadsheet. It gives abbreviated life tables for the eight potential clients. Look at the third column (Status) for each potential client. For each client, the rest of his or her life has been randomly simulated based on the probabilities that he or she lives through each year. Jacob, for example, is a newborn, so during each of the age ranges listed, it's possible that he'll die, but the probabilities of dying for the age ranges at the beginning of his life are small.

Each time you press F9 on your keyboard, you see an alternate life for Jacob, with his status for each age range shown as either alive or dead. If the dead were first to appear for the age range of 75 to 76, for example, this would mean that Jacob died between the ages of 75 and 76, or that he lived to be 75 years old.

a.Press F9 on your keyboard a few times and see how long Jacob lives in each of his alternate lives. How long did Jacob live each time?

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The rest of the potential clients are similar to Jacob, but since they've already lived parts of their lives, their status will always be alive for the age ranges that they've already lived.

For example, Carol is 44 years old, so no matter how many times you press F9 on your keyboard, Carol's status will always be alive for all the age ranges up to 43-44. Starting with the age range of 44-45, however, there is the possibility that Carol's status will be dead.

b.Press F9 on your keyboard five more times and see how long Carol lives in each of her alternate lives. Remember that she will always live to be at least 44 years old, since she is already 44 years old. How long did Carol live each time?

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Now you will find the percent survival of each of your eight clients to the end of his or her policy using the simulation in the spreadsheet.

For each potential client, you will see whether he or she would be alive at the end of his or her policy. The cells in the spreadsheet that you should look at to determine this are highlighted in yellow.

c.Next, go to the worksheet labeled Task 2b and record either alive or dead for the first trial. Once you complete this, the All column will say yes if all the clients were alive at the end of their policies or no if all the clients were not alive at the end of their policies. Were all the clients alive at the end of their policies in the first trial?

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Next, go back to the Task 2a worksheet, press F9, and repeat this process until you have recorded 20 trials in the Task 2b worksheet.

In the Percent Survived row at the bottom of the table on the Task 2b worksheet, it will show the percentage of times each client survived to the end of his or her policy, and it will also show the percentage of times that all of the clients survived to the end of their respective policies. Check to see whether these percentages are in line with the probabilities that you calculated in questions 1 through 9 in Task 1.

d.Are your probabilities from the simulation close to the probabilities you originally calculated?

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Task 3: Randomly Choosing Two Potential Clients

For this task, you'll use the same set of eight potential clients: Jacob, Carol, Geraldo, Meg, Earvin, Dora, Adam, and Sally. You have decided to just randomly choose two of them to offer life insurance policies.

a.How many pairs of potential clients can be randomly chosen from the pool of eight candidates?

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b.What is the probability of any particular pair being chosen?

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c.What is the probability that the pair chosen is Jacob and Meg or Geraldo and Sally?

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d.What is the probability that the pair chosen is Carol and Earvin, Earvin and Dora, or Dora and Adam?

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e.What is the sample space of the pairs of potential clients that could be chosen?

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References: Life tables https://www.cdc.gov/nchs/data/nvsr/nvsr59/nvsr59_09.pdf

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