Math / Data & Statistics

QuestionThere is a 0.9982 probability that a randomly selected 30 -year-old male lives through the year. A life insurance company charges $\$ 187$ for insuring that the male will liv through the year. If the male does not survive the year, the policy pays out $\$ 100,000$ as a death benefit. Complete parts (a) through (c) below. a. From the perspective of the 30 -year-old male, what are the monetary values corresponding to the two events of surviving the year and not surviving?
The value corresponding to surviving the year is $\$-187$ . The value corresponding to not surviving the year is $\$ 99,813$ . (Type integers or decimals. Do not round.) b. If the 30-year-old male purchases the policy, what is his expected value?
The expected value is $\$-7$ . (Round to the nearest cent as needed.) c. Can the insurance company expect to make a profit from many such policies? Why?
Yes, because the insurance company expects to make an average profit of $\$ \square$ on every 30 -year-old male it insures for 1 year. (Round to the nearest cent as needed.)

Studdy Solution

STEP 1

What is this asking? This problem wants us to figure out the potential outcomes for a 30-year-old male buying life insurance, see what he can expect to "gain" on average, and determine if the insurance company makes a profit. Watch out! Don't mix up the probabilities of surviving and not surviving!
Also, remember that the cost of the insurance is a loss for the male, so we'll represent that with a negative value.

STEP 2

1. Calculate the Value of Not Surviving
2. Calculate the Expected Value for the Male
3. Calculate the Expected Profit for the Insurance Company

STEP 3

Let's define what "value" means in this context.
It's the net monetary gain or loss for the male.

STEP 4

If the male doesn't survive, the policy pays out $\$100,000$ .
But he did pay $\$187$ for the policy.
So, the net value is the payout minus the cost:
$\$100,000 - \$187 = \$99,813$

STEP 5

The expected value is the average outcome if the same scenario played out many times.
It's like a weighted average of the possible outcomes.

STEP 6

We multiply each outcome by its probability and add those products together.
The probability of surviving is $0.9982$ , and the value of surviving is $-\$187$ (because it's a cost).
The probability of not surviving is $1 - 0.9982 = 0.0018$ , and the value of not surviving is $\$99,813$ , as we calculated earlier.

STEP 7

So, the expected value is:
$(0.9982) \cdot (-\$187) + (0.0018) \cdot (\$99,813)$ $= -\$186.6734 + \$179.6634$ $= -\$7.01$ This means, on average, the male would lose about $\$7$ .

STEP 8

The insurance company's profit is the opposite of the male's gain/loss.
If the male survives, the company gains the $\$187$ premium.
If the male doesn't survive, the company loses $\$99,813$ (after accounting for the premium they already received).

STEP 9

The expected profit for the insurance company is calculated similarly to the expected value for the male:
$(0.9982) \cdot (\$187) + (0.0018) \cdot (-\$99,813)$ $= \$186.6734 - \$179.6634$ $= \$7.01$

STEP 10

a. Surviving: $-\$187$ ; Not surviving: $\$99,813$ b. Expected value for the male: $-\$7.01$ c. Yes, the insurance company can expect to make a profit.
They expect to make an average profit of $\$7.01$ on every 30-year-old male they insure for one year.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the Value of Not Surviving
2. Calculate the Expected Value for the Male
3. Calculate the Expected Profit for the Insurance Company

STEP 3

Let's define what "value" means in this context.
It's the net monetary gain or loss for the male.

STEP 4

If the male doesn't survive, the policy pays out $\$100,000$ .
But he did pay $\$187$ for the policy.
So, the net value is the payout minus the cost:
$\$100,000 - \$187 = \$99,813$

STEP 5

The expected value is the average outcome if the same scenario played out many times.
It's like a weighted average of the possible outcomes.

STEP 6

STEP 7

So, the expected value is:
$(0.9982) \cdot (-\$187) + (0.0018) \cdot (\$99,813)$ $= -\$186.6734 + \$179.6634$ $= -\$7.01$ This means, on average, the male would lose about $\$7$ .

STEP 8

The insurance company's profit is the opposite of the male's gain/loss.
If the male survives, the company gains the $\$187$ premium.
If the male doesn't survive, the company loses $\$99,813$ (after accounting for the premium they already received).

STEP 9

The expected profit for the insurance company is calculated similarly to the expected value for the male:
$(0.9982) \cdot (\$187) + (0.0018) \cdot (-\$99,813)$ $= \$186.6734 - \$179.6634$ $= \$7.01$

STEP 10

a. Surviving: $-\$187$ ; Not surviving: $\$99,813$ b. Expected value for the male: $-\$7.01$ c. Yes, the insurance company can expect to make a profit.
They expect to make an average profit of $\$7.01$ on every 30-year-old male they insure for one year.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the Value of Not Surviving2. Calculate the Expected Value for the Male3. Calculate the Expected Profit for the Insurance Company

STEP 3

Let's **define** what "value" means in this context.It's the net monetary gain or loss for the male.

STEP 4

If the male doesn't survive, the policy pays out $100,000\$100,000$100,000.But he *did* pay $187\$187$187 for the policy.So, the **net value** is the payout *minus* the cost:$100,000−$187=$99,813 \$100,000 - \$187 = \$99,813 $100,000−$187=$99,813

STEP 5

The **expected value** is the average outcome if the same scenario played out many times.It's like a weighted average of the possible outcomes.

STEP 6

STEP 7

STEP 8

The insurance company's profit is the opposite of the male's gain/loss.If the male survives, the company gains the $187\$187$187 premium.If the male doesn't survive, the company loses $99,813\$99,813$99,813 (after accounting for the premium they already received).

STEP 9

STEP 10

a. Surviving: −$187-\$187−$187; Not surviving: $99,813\$99,813$99,813 b. Expected value for the male: −$7.01-\$7.01−$7.01 c. Yes, the insurance company can expect to make a profit.They expect to make an average profit of $7.01\$7.01$7.01 on every 30-year-old male they insure for one year.

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1. Calculate the Value of Not Surviving
2. Calculate the Expected Value for the Male
3. Calculate the Expected Profit for the Insurance Company

Let's define what "value" means in this context.
It's the net monetary gain or loss for the male.

If the male doesn't survive, the policy pays out $\$100,000$ .
But he did pay $\$187$ for the policy.
So, the net value is the payout minus the cost:
$\$100,000 - \$187 = \$99,813$

The expected value is the average outcome if the same scenario played out many times.
It's like a weighted average of the possible outcomes.

The insurance company's profit is the opposite of the male's gain/loss.
If the male survives, the company gains the $\$187$ premium.
If the male doesn't survive, the company loses $\$99,813$ (after accounting for the premium they already received).

a. Surviving: $-\$187$ ; Not surviving: $\$99,813$ b. Expected value for the male: $-\$7.01$ c. Yes, the insurance company can expect to make a profit.
They expect to make an average profit of $\$7.01$ on every 30-year-old male they insure for one year.