EXTRA CREDIT PROBLEMS FOR WD SMTIH, Math55(31). HAND IN AT ANY TIME, TO GET EXTRA CREDIT. PLEASE CLEARLY LABEL AND PUT ON OTHER SHEETS FROM REGULAR HW. (these are harder than regular problems and you may have to write a little essay to show me how you are doing it... in some cases you may have to get data from books or the internet to use... these are mostly NOT plug and chug problems - they require you to think up a plan and execute it - more like real world math problems and less like textbook exercises...) ------------------------------------- YOU GET TO PICK WHICH OF THESE YOU WANT TO TRY. --------------------------------------------- MUST SHOW ALL WORK. IF GOT HELP, MUST GIVE EXPLICIT NAMED CREDIT TO ALL HELPERS, BOOKS, WEB SITES, OR WHATEVER. ----------------------------------------------- 1. You are dealt 5 cards from a standard 52-card pack. What is the probability you get these 5 cards A K Q J 10 (any combination of suits ok, and not necessarily coming to you in this time-order)? 2. You roll 3 dice (red, green & blue). What is the probability that they add up to 5? 3. You are dealt 5 cards from a standard 52-card pack. What is the probability you get 4 aces among those 5 cards? 4. There are N people in a room. What is the smallest N so that it is a better-than-even bet (I.e. prob > 1/2) that two, among those N, people have the same birthday? (Assume 365 days per year and all those 365 days equally likely.) 5. You are dealt 5 cards from a standard 52-card pack. What is the probability you get 2 pairs among those 5 cards? (E.g. you get 2,2, 4,4, J which would be a pair of 2s and a pair of 4s.) 6. It has been said that "total number of deaths" is not the best way to measure the severity of some kind of problem (like pollution). A better way is total "number of years of lost life." I.e. if you die at age 10 that is more severe than if you die at age 90. Try to make a table of 10 of the most important causes of death by this second measure - for each list (a) the cause, and (b) the number of years (or days, or hours, if more appropriate) of lost life that cause is expected to cost the average US citizen (c) how you arrived at the number or estimate. Also include in your table these 2 recent newsworthy things: * sidestream tobacco smoking * terrorism-caused 7. Estimate: how many hairs do you have? 8. Estimate: how many atoms in your body? 9. Estimate: How many liters of air do you inhale in 1 year? That would be a cube of air how many meters on each side? 10. Estimate: How many liters of water do you drink (include coke, pepsi, beer, etc.) in 1 year? That would be a cube of liquid how many meters on each side? 11. (related to 9). How many years would it take for the human race to breathe all the air on the planet? (Assume for simplicity that world population stays at 6.5 billion and that air never got refreshed by photosynthetic plants, and every chunk of air got breathed exactly once and never again.) 12. (related to 10). How many years would it take for the human race to drink all the ocean water on the planet? (Assume for simplicity that world population stays at 6.5 billion and that water never got recycled... and that the salt didn't bother us...) 13. Somebody pours a 1-liter bottle of arsenic atoms into a river. Assume that arsenic gets completely mixed into all the water in the world. How many atoms of that arsenic will be in your next 1-liter drink of water? 14. What is the probability that the next 5 random people you see, will have birthdays on 5 consecutive days (not necessarily in the same order as the order you meet the 5 people)? 15. I saw the TU basketball team play a game, and they seem to me to have some mathematical problems! (Though David Hawkins - in our class and on that team - seems to me, individually, to be pretty damn good!) Try to analyze the following basketball-strategy problem... (To really solve it well, you'd have to find known statistics of what % of the time 3-point shots miss, and what % of the time a team ball-possession leads to a basket for them, and what % of all baskets are 3-points, what % 2-points, and what % 4-point plays... and what % of misses get rebounded by the defensive team vs the offensive team... and what % of ball-possessions get turned over by errors before the team even gets to shoot... so its quite hard to really do it precisely... but you can aprpoximate...) (a) Suppose TU takes a 3-point shot when the enemy team has got the TU players completely boxed out of the area near the basket, so if it misses, the enemy will almost certainly get the rebound. (If hits, TU gets 3 points.) (b) Same thing, but the TU players are positioned so that they have a 33.3% chance of getting the rebound (if is one). QUESTION: Obviously (b) is better than (a) - but how much better? - what is the expected lost number of points for TU every time they choose to go with (a) instead of (b)? 16. You have a choice of renting an apartment for $1000/month rent, or buying it for $100,000 then paying "fees" of $300/month from then on. Rents rise 5% per year. So do housing prices(and you expect to be able to resell at the future price). You have the $100,000 in cash now. You can deposit it in a 7%-APR account instead of spending it, though. Buying will also cost you $2000 in one-time "costs". - ok, which is the better choice - renting or buying? Explain why you think so. Does the best choice change depending on the timespan you plan to live there? If so, what timespan is the critical one that causes the changeover? 17. Suppose your life expectancy is 10 more years. What can you deduce about the probability that you will, in 20 years, be dead? (What is the most that probability could be, and what is the least it could be?) 18. If being black has nothing to do with the chance you are a good basketball player and nothing to do with your choice of career, then what would the probability be that some NBA team would pick (as its next 10 draft picks, assuming unrealistically that all draft choices independent) all blacks? (You'll have to find out the percentage of blacks in the US population...)