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### Course: Statistics and probability > Unit 9

Lesson 8: More on expected value- Term life insurance and death probability
- Getting data from expected value
- Expected profit from lottery ticket
- Expected value while fishing
- Comparing insurance with expected value
- Expected value with empirical probabilities
- Expected value with calculated probabilities
- Making decisions with expected values
- Law of large numbers

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# Comparing insurance with expected value

Sal uses expected value to compare a couple of different insurance policies. Created by Sal Khan.

## Want to join the conversation?

- Can you update this example and use better numbers for the medical costs? I am getting an expected value of $3200 for medical expenses ... and it looks like Paul would be much better off self-insuring. Maybe make the last event have a cost of $150,000 instead of $15,000??(20 votes)
- Maybe that is an easter egg Sal hid for us curious learners to compute by ourselves and see how the world is a cruel and deceptive place to live in?(33 votes)

- I still don't understand why we aren't adding 8000 into all of the costs. E.g. 8000 * 0.3 + 9000 * 0.25 ..... . I understand that the 8000 is a cost no matter what, but when we are doing similar problems regard an arcade game that costs $2 to play and but gives a $40 reward for winning, we subtract 2 from both the winning and losing possibilities. In that case, you're spending the 2 dollars no matter if you win or lose. So why do we account for the "fee" in both outcomes for the arcade game, but not account for the "fee" in all outcomes with the insurance?(4 votes)
- It doesn't make any difference to the result whichever of those ways you calculate it. Sal is taking advantage of the fact that E(X + k) = E(X) + k, where X is the random variable (the medical costs) and k is a constant (the price of the insurance).(9 votes)

- At3:45, why do we add $
**8000**to calculate the expected medical costs with the low deductible plan, instead of $**9000**? Can anyone explain it to me more clearly?(4 votes)- $8,000 is your annual insurance bill (what you pay the insurance company for the benefits of the $1000 deductible). Think of it as your membership fee, totally independent of any actual medical costs.(4 votes)

- Not going with any insurance plane seems to be the best, it has expected cost of 3,200 dollars(4 votes)
- lol That might be the best choice.

Also, there are alternative ways to insurance like Samaritan which may be a better way.(0 votes)

- This is not a correct cost benefit analysis in my opinion. Payoff should be calculated in terms of net cost which includes what net benefit he derives from insurance.

expected value from first plan should be calculated in terms of net benefit--

Plan 1--

0.3*(-8000)+(-8000-1000)*0.25+(-8000+4000-1000)*0.2+(-8000+7000-1000)*0.2+ (-8000+15000-1000)*0.05

that turns out to be -5500. so, from first plan he actually will lose out -5750 per year on an average.

Plan 2:

E(benefit) = (-7500)*0.3 + (-7500-1000)*0.25 +(-7500+4000-2500)*0.2+(-7500+7000-2500)*0.2+(-7500+15000-2500)*0.05 = -5925

so, if someone forces him to buy one of the plan, he is better off buying first plan. what do you think?(3 votes)- Your analysis is correct. When comparing insurance plans, it's crucial to consider the net cost, which includes both the cost of the insurance premium and the expected out-of-pocket expenses for medical costs.

Let's go over your calculations:

Plan 1 (Low-deductible plan):

The net benefit can be calculated as follows:

E(benefit) = 0.3 × (−8000) + 0.25 × (−8000 − 1000) + 0.2 × (−8000 + 4000 − 1000) + 0.2 × (−8000 + 7000 − 1000) + 0.05 × (−8000 + 15000 − 1000)

E(benefit) = −5500

So, on average, Paul would lose $5500 per year with Plan 1.

Plan 2 (High-deductible plan):

Similarly, for Plan 2: E(benefit) = 0.3 × (−7500) + 0.25 × (−7500 − 1000) + 0.2 × (−7500 + 4000 − 2500) + 0.2 × (−7500 + 7000 − 2500) + 0.05 × (−7500 + 15000 − 2500)

E(benefit) = −5925

With Plan 2, on average, Paul would lose $5925 per year.

Given these calculations, if Paul had to choose between the two plans, he would be better off purchasing Plan 1 (the low-deductible plan), as it results in a lower expected net cost compared to Plan 2.(1 vote)

- How come concept is explained later but a question on this concept was given in the previous exercise which ofcourse I was not able to score. :((1 vote)
- What's the difference of calculating the expected value like Sal does it versus calculating the expected value of the medical cost distribution (3.2k USD) and then using that as the average cost per year and stating that on average he'll pay 9k for the low plan and 10k for the high plan? (Average cost per year exceeds both 1k and 2.5k so he has to pay max in either plan's case, so then you'd just add the two)

What am I calculating here and why is it wrong? I still get the same result in the sense that the low plan is better but it's also immediately evident that you should just opt to pay your bills without insurance since you're average annual cost will be 3.2k anyway -- unless my math is fundamentally broken.(1 vote)- When I tackled the problem before seeing how Sal did it, I also did it the way you did it - determining that statistically the cost would be $3200 per year for this person. Our way and Sal's way both make sense to me yet the math is different, and that is a troublesome situation indeed!(1 vote)

- The video is great as always, but this example of a "health insurance" is the epitome of America... as someone from a country with an actual and affordable health unsurance, these numbers seem insane. Why would anyone even get a "health insurance" like this if you pay them but basically you get almost nothing back at all like Sal showed in the alculations.(1 vote)
- Hey there's a quiz question in there that's insisting that a low deductible insurance plan with a cost of E(x) = $6,500 is a better choice than high deductible plan with a cost of E(x) = $5,540? Wut?(1 vote)
- If the quiz question states that the low-deductible plan with a higher expected cost is a better choice than the high-deductible plan with a lower expected cost, it would be inconsistent with the principles of cost-benefit analysis. It's essential to choose the plan with the lower expected net cost, which may not always align with the plan with the lower premium cost.(1 vote)

- I can't understand why are we multiplying percentage and costs. It somehow doesn't make sense, what product of probability and medical costs represents?(0 votes)
- It is probability that how much of the cost you will pay. This is how expected value thing works. P(thing) times cost of things is probability or expected value that you will end up paying.(1 vote)

## Video transcript

Voiceover:Paul has the option
between a high-deductible or a low-deductible health insurance plan. When we talk about the
deductible in health insurance, if someone says that they have a plan with $1,000 deductible, that means that the insurance company only pays the medical cost
after the first $1,000. So if you have a $1,000 deductible and you, say, incur medical cost of $800, you're going to pay that $800. The insurance company won't pay anything. If you have a deductible of $1,000 and your total medical costs are $1,200, you're going to pay the first $1,000, and then the insurance company
will kick in after that. So with that out of the way, let's think about his 2 plans. If Paul chooses the low-deductible plan, he will have to pay the first $1,000 of any ... let me do that in purple, the first $1,000 of any medical cost. The low-deductible plan
costs $8,000 for a year. So in this situation,
he's going to pay $8,000 to get the insurance. If he has $900 of medical expenses, the insurance company still pays nothing. If he has $2,000 of medical expenses, then he pays the first $1,000, and then the insurance company would pay the next $1,000. If he has $10,000 in medical expenses, he would pay the first $1,000, and then the insurance company would pay the next $9,000. If Paul chooses the high-deductible plan, he will have to pay the first $2,500 of any medical cost. The high-deductible plan
costs $7,500 a year. It makes sense that the
high-deductible plan costs less than the low-deductible plan because here, the
insurance doesn't kick in until he has over $2,500
of medical expenses, while here, it was only $1,000. To help himself choose a plan, Paul found some statistics
about common health problems for people similar to him. Assume that the table below, and I put it up here on the right, correctly shows the probabilities in cost of total medical incidents
within the next year. So this right over here, it's saying what's the probability he
has $0 in medical costs? What's the probability of ... he has a 25% probability of $1,000, 20% probability, $4,000. And this is a simplification, a pretty dramatic simplification
from the real world. In the real world, the
way this makes it sound is there's only 5 possible medical costs that someone might have, $0, $1,000, $4,000, $7,000, and $15,000. In the real world, you could
have $20 of medical cost. You could have $20,000. You could have $999. So in the real world,
there is many, many more situations here that you
would have to redistribute the probabilities accordingly. But with that said, this
isn't a bad approximation. It's just saying, OK, roughly, if you wanted to construct this so it's easier to do the math, which the problem writers have done, say, look, OK, 30% $0, 25% $1,000. This is pretty indicative if you had to group all
of the possible costs into some major bucket. So it's probably at least
a pretty good method for figuring out which insurance policy someone should use. So they say, including
the cost of insurance, what are Paul's expected
total medical costs with the low-deductible plan? Round your answer to the nearest cent. Actually, I'll get the
calculator out for this. With the low-deductible plan here, low-deductible plan, he's going to have to
spend his total cost, he's going to spend $8,000 no matter what. Whoops, what happened to my calculator? He's going to spend $8,000 ... My God, I'm having issues. He's going to spend $8,000 no matter what, so that is $8,000. Then let's see. There is a 30% probability
he spends nothing. I could just write that
as plus 0.3 times $0, and I will write it just so that you see I'm taking that into account. There's a 25% chance that he
has $1,000 in medical costs. In the low-deductible plan, he still has to pay that $1,000, so plus 0.25% chance that he pays $1,000, $1,000. And then you might say,
OK, plus 0.2 times $4,000, but remember, if his
medical costs are $4,000, he is not going to pay that $4,000. He is only going to pay the first $1,000. So it's really plus 0.2. In this situation, his
out-of-pocket costs are only $1,000. The insurance company
will pay the next $3,000. So 0.2 times $1,000. And then plus 0.2, 20% chance, even if he has $7,000 in medical costs, he's only going to have
to pay the first $1,000, so 0.2 times $1,000 again. And then plus 0.05 times,
once again, $1,000. If he has $15,000 in expenses, he is only going to have
to pay the first $1,000, times $1,000. And we get $8,700. One way you could have thought about it is OK, he is going to have
to pay $8,000 no matter what, and all of the situations
where he ends up paying, that's anything, that's these 4 situations right over here, there is a 70% probability of falling into one of these 4 situations. At any one of these, he
only has to pay $1,000. The insurance company pays
everything after that. So you could say $8,000
plus there is a 70% chance that he is going to pay $1,000. And once again, this table is
a pretty big simplification from the real world. There's probably a lot of scenarios where you would have to pay $500 or $600 or whatever it might be. But let's just go with this. That's essentially a simplification. There's a 70% chance that he's
going to have to pay $1,000, and so that's $700
expected cost from that, plus the $8,000 from the insurance gets us to $8,700. So let's write that down. So $8,700 ... My pen is really acting up. I don't know what's going on here. I think I have to get a new tablet. You can't even read that. Let me write this. $8,700. Including the cost of insurance, what are Paul's expected
total medical costs with the high-deductible plan? Round your answer to the nearest cent. Let's look at the high-deductible plan. He's going to pay $7,500 no matter what, $7,500 no matter what, and then there's a $0 ... We could write $0 times
30% or 30% times $0, but that's just going to be $0. There is a 25% chance he spends $1,000, so plus 0.25 times $1,000. There is a 20% chance, plus 0.2, he's not going to spend $4,000 here. He's going to have to
spend the first $2,500. So it's a 20% chance he spends $2,500, so times $2,500. Insurance company will
pay the next $1,500. Plus another 20% chance. Even in this situation,
he only has to pay $2,500, so times $2,500, $2,500. And then finally, plus
there's a 5% chance. Even in this situation, he only
has to pay the first $2,500, times $2,500 gets us to $8,875. So $8,875. Once again, you could think about it as, OK, there is a 25%
chance that he pays $1,000, and then there is a 45%
chance that he pays $2,500. All of these situations,
he is paying $2,500. But either way, you would get $8,875. If Paul wants the best
payoff in the long run and must buy 1 of the 2 insurance plans, he should purchase the? Well, his expected
total cost of insurance, including medical costs, is lower with the low-deductible plan. So this one, he should go with the low, low-deductible. Which, once again, you shouldn't use these
videos as insurance advice. This is actually ... But also, it's an interesting
way to think about it. It's typically ... Well, it's not always typically the case that the low-deductible
plan is going to have a higher long-term ... or the low-deductible plan
is going to be a better deal. It's usually the ... Well, I won't make any
actuarial statements. But at least in this situation, the low-deductible plan
seems like the better deal. He has lower expected total
costs given these probabilities.