Batch 3 - Class 185 - Probability in Insurance

• There are four togglers (a Toggler alternately tells the truth and lies) and one truthteller in a room. Ask two question (both to same person, or one each to different people) to determine who is the truthteller.
• Answer: Ask to same guy, "Are you the truthteller", "Who is the truthteller" if Answer is Yes, and "Who is not a truthteller" if Answer is No. See video for more explanation - https://www.khanacademy.org/math/recreational-math/brain-teasers/v/toggler-brain-teaser

• Probability of a dice roll resulting in number 3 (Answer: 1/6)
• Why? Kids are likely to come up with "one face out of 6"
• Verify by rolling the dice 30 times - emphasize that probability problems are essentially counting problems
• Get kids to take examples where one can do probability as in above case
• Independent Events - If we have rolled the dice 5 times and we get 5 1's, what are the chances of getting a 1 on next throw? How about chances of getting a 2?
• Simultaneous Independent Events - Coin tosses - Chances of heads, Chances of one head and one tail in two throws, Chances of first head and second tail in two throws
• Expected Value - Explain the notion - Lets say there is a lottery ticket, where if you match the last digit, you get Rs 100, if you match last 2 digits, you get Rs 5000 and if you match all three digits, you get Rs 50000. What is the expected value of winnings?

• A person is 55 years old, and wants to make sure that he has enough money when he is 70. So he decides to put 1 million rupees every year into a saving bank account, which pays him 5% interest each year on compounded basis. Hence the money is constantly multiplying and he will get this when he turns 70. However, there is a 20% chance that he dies before turning 70 and the money is lost. What is his expected value?
• First step - calculate how much will he get if he survives = 1*(1+5%)^15 with compounding, i.e. 2.08 million (call this S). This will happen with 80% probability
• If he dies before 70, he gets zero
• So expected value = 0.8*S
• What is the bank's expected value here? 
• At end of 15 years, 0.2*S (even if the person dies earlier, we can assume that bank keeps accruing interest till 15 years)
• However, its two very distinct possibilities - to earn zero or to earn S

• The person now finds two more people exactly like him. They decide that at age of 70, the surviving folks will split the total of remaining balance. What happens in this case? What is the expected value?
• What are the different scenarios, the probabilities of those scenarios and expected value for the person
• The person now expects to get 0.992*S - and each of the other people do too
• This seems like a much better scheme for everyone
• Why? Because the chance that no one gets the money is small (when everyone dies)
• Is it better for the bank - actually from a risk standpoint, yes. Because the chances that the bank will loose money is very small.

• What would you expect if a hundred people got together for the mutual scheme? A thousand?
• The expected value will start to approach S
• This "pooling of risk" is the fundamental concept behind insurance as well
• Segue into binomial distributions

• MartinShColl 2.4 - Three points are selected at random on a sphere's surface. What is the probability that they fall within the same hemisphere?
• Answer: 1 (It is a certainty)