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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.

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• 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??
• 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?
• 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?
• 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).
• At , 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?
• \$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.
• Not going with any insurance plane seems to be the best, it has expected cost of 3,200 dollars
• lol That might be the best choice.

Also, there are alternative ways to insurance like Samaritan which may be a better way.
• 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?
• 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.

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?