EXTRA CREDIT PROBLEMS FOR MATH 55/31 WD SMITH *SOME ANSWERS* ============================================================ One thing I don't like about this (math 55) course is, it promulgates the bad attitude that what math is about, is just memorizing formulas and plugging numbers into them. That approach actually works OK for the financial stuff in the first half of the course -- because, let's face it, there are not too many financial questions people want to ask -- but it does NOT work well for math in general. Because there are a HUGE huge number of possible kinds of math problems and questions people could want to ask. So it is impossible to memorize a formula for every one of them. Instead you have to take the attitude that you have to glue together basic components - basic rules, basic tools - in the right way to solve a problem. It is less robotic. It involves more thinking for yourself, moe creativity. So I was glad to see in some of these extra credit problems, some of the students definitely going in that direction. -wds ------------------------------------------------------------ 2. What is the probability 3 dice rolls will add up to 5? ANS. 6 ways to add to 5 namely 113 131 311 122 212 221 and 6*6*6=216 ways total, so prob=6/216=1/36. 3. If dealt 5 cards from a 52-card pack, what is the probability you get 4 aces (plus something else)? ANS. There are 52C5=2598960 hands. There are 48 possible hands with 4 aces (AAAAx where x can be 48 possible cards). So prob=48/2598960=1/54145. 4. N people in a room, what is the smallest N such that the chances are better than 50-50 that 2 of them have the same birthday? ANS. Nichole Morrill found a web site about this very problem! www.mste.uiuc.edu/reese/birthday/explanation !! 365*364*363*...*344*343 Prob(2 Bdays same) = 1 - ------------------------- 365*365*365*...*365*365 if there are 23 people (note 23 terms on the top, and 23 on the bottom of the fraction) which slightly exceeds 0.5, so the answer is N=23. (Surprisingly small number of people needed.) 5. If dealt 5 cards from 52, what is the probability you get 2 pairs? ANS. If we count 22 44 J as a pair of 2s and a pair of 4s, but 22 444 NOT as two pairs, and 2222J NOT as two pairs (to make the rules precise) then the number of hands with 2 pairs are 13C2 * 4C2 * 4C2 * 44 = 123552 since 13C2 possible choices of 2 kinds of cards, 4C2 possible choices for a pair of 3s (once you've decided it is going to be 3s) since 4 suits, and 48 choices for the remaining unpaired 5th card. Comparing with 52C5=2598960 possible hands, we have probability = 123552/2598960 = 1/21.035, which is surprisingly probable. 7. How many hairs do you have? ANS. Nichole Morrill found a web site on this too... www.dermatology.org/skincare/hair/hairfact.html She estimates 100,000 head hairs and hence 15,000,000 total hairs on whole body, figuring her body is like 15 heads. BUT maybe there are a lot fewer hairs per square cm on your body than there are on your head? I tried counting hairs on 1 square cm of my arm and it seems like about 10-20 per square cm. That extrapolates to only about 10000-20000 body hairs, in which case your head is much hairier than the rest of your body. 8. How many atoms in body? ANS. 23 There are 6.02*10 atoms in a "mole". Most of your body is water. A mole of water is 18 grams (16 for the O and 2 for the H2). 26 If you weigh 90 kg (fairly heavy) that is 500 moles which is 9*10 atoms taking into account the fact that a water molecule is made of 3 atoms. (Note: some people said MUCH huger numbers, larger than the number of atoms in the whole galaxy! You have to do some sanity checking...) 9. Nichole Morrill found a web site to tell you your typical lung capacity... she says about 4.5 liters (deep breath), so she estimates 2 liters regular breath. So breathing 10 times per min leads to 60*24*365*2*10=1051200 liters breathed per year. This is 1051200000 cubic cm per year (1 liter=1000 cubic cm). Take that to the power 1/3 to get the cube sidelength of about 1017 cm, i.e. each year you breathe a cube of air about 10.2 meters on a side (weighing about 1 ton). 10. If you drink 2 liters per day, that in 1 year is 730=365*2 liters which is 730,000 cubic cm. The cube root of 730,000 (i.e. 730,000 to the power 1/3) is about 90. So each year you drink a cube of water 90cm = 0.9 meter on a side. (This also weighs about 1 ton. Also, the total amount of food you eat per year is about 1 ton, too.) 11. The total air mass on the planet is about 1 kg per square cm of Earth surface (typical barometric pressure = the weight of the air above each square cm pressing down on that square cm). So the surface area of a sphere of radius r is 2 4*pi*r and the Earth radius r is about 5000 km (I did not look this up so don't trust it too much), so... the total mass of the air is about 6 10 18 4*3.1416*5*5 * 10 * 10 kg = 3.2 * 10 kg = 3.2 million billion tons (1 metric ton = 1000 kg, 1 km is 100000 cm). So, the human race (6.5 billion people) would breathe this much air (at 1 ton per person per year) in about 0.5 million years. (And there are other ways to estimate the total amount of air on the Earth besides this one.) 12. If there are 6.5 billion people each drinking 730 liters per year, and estimating the volume of the ocean as 21 1.25 * 10 liters (Nichole Morrill found that number on yet another web site...) 8 it would take 2.6 * 10 years for the human race to drink the oceans. That's 260 million years. As a sanity check on Nichole's ocean volume, figure the ocean is 5 km deep and maybe 10000 km wide, leading to volume 5000*5000*pi*5 cubic km, which is 390 million cubic km. As liters that would be 6 12 390 * 10 * 10 which is 20 3.9 * 10 liters, which is 3 times less than her figure, so anyway, her figure is not totally insane so I'm willing to believe it. 13. How many atoms of Arsenic will be in your next drink of water, if a 1 liter bottle of arsenic were mixed into all the water in the world? ANS. If there were 50 moles of arsenic in the 1 liter bottle (roughly right) then that would be 25 10 arsenic atoms. 21 Mixing your 1 liter into all the 1.25 * 10 liters of water in the oceans would be 8000 arsenic atoms per liter. (Actually you probably drink a lot more arsenic than that.) 14. What is the probability the next 5 random people you see will have birthdays on 5 consecutive days (but not necessarily in the same order you see them)? ANS. There are 365 choices for the earliest of the 5 birthdays and the rest are then determined. There are 365*365*365*365*365 ways the 5 Bdays could have come out. So the probability is 365/(365*365*365*365*365) = 1/(365*365*365*365) = 1/17,748,900,625 (ignoring leap years...).