Probability, Genetics, and Games

Transcription

1 " Probability, Gntics, and Gams Hav you vr hard of gns? (W don t man th kind you war!) What color ar your ys? Can you curl your tongu? Your birth parnts gav you a uniqu st of gns that dtrmin such things. Scintists who study traits such as y and hair color ar calld gnticists (juh NET uh sists). Gnticists us probability to prdict crtain traits in childrn basd on traits in thir parnts or rlativs. 4.1 Gntic Traits Look at th arlob of a classmat. Is it attachd or dos it dangl frly? Th typ of arlob you hav is a trait dtrmind by your gns. Hr is a dscription of four gntic traits: Attachd arlob: An arlob is attachd if its lowst point is attachd dirctly to th had, as shown blow. Dimpl: A dimpl is a small indntation, usually nar th mouth. Straight hair: Straight hair has no wavs or curls. (Not: Considr only how a prson s hair is naturally.) Widow s pak: A widow s pak is a V-shapd hairlin, as shown blow. Attachd arlob Unattachd arlob Widow s pak No widow s pak Invstigation 4 Probability, Gntics, and Gams 55

2 Problm 4.1 Applying Exprimntal Probability Th tabl lists four gntic traits. Classroom Gntics Survy Trait Attachd Earlobs Dimpls Straight Hair Widow s Pak Ys No Total A. Copy th tabl. Find th numbr of popl in your class who hav ach trait and rcord th rsults in your tabl. B. Us your tabl to complt parts (1) (4). 1. For ach trait, find th probability that a prson chosn at random has th trait. 2. What is th probability that a prson chosn at random dos not hav straight hair? 3. How many studnts in your school do you xpct to hav attachd arlobs? 4. How many studnts in your school do you xpct to hav a widow s pak? C. Blow ar th rsults of a study of studnts from around th country. U.S. Gntics Survy Trait Attachd Earlobs Dimpls Straight Hair WidowÕs Pak Ys No 1,080 1, Find th probability that a prson chosn at random has ach trait. 2. How do th probabilitis in Qustion B compar to th probabilitis from th national data? Homwork starts on pag How Likly Is It?

3 4.2 Tracing Traits In th last problm, you lookd at xprimntal probabilitis for crtain traits. In som cass, you can dtrmin th probability that a child will hav a trait basd on his or hr parnts gns. Gnticists us th word alll (uh LEEL) for on of a pair of gns that dtrmins a trait. For xampl, you hav two allls that dtrmin whthr your arlobs ar attachd. You rciv on of ths allls from your birth mothr and on from your birth fathr. Of cours, ach parnt has two arlob allls. Lt s us to rprsnt th alll for attachd arlobs. Lt E rprsnt th alll for nonattachd arlobs. If you rciv an alll from ach parnt, your arlob allls will b, and you will hav attachd arlobs. If you rciv an E alll from ach parnt, your arlob allls will b EE. Thn you will hav nonattachd arlobs. What if you rciv on E and on alll? In natur, th E alll is dominant and th alll is rcssiv. This mans that you hav an E combination, th E dominats, and you will hav nonattachd arlobs. Earlob Allls Lttrs EE E or E Earlob Trait Nonattachd Nonattachd Attachd Invstigation 4 Probability, Gntics, and Gams 57

4 An Exampl: Bonni and Evan s Baby Bonni and Evan ar going to hav a baby. Bonni s arlob allls ar E, and Evan s arlob allls ar. You can dtrmin th probability that thir baby will hav attachd arlobs by making a tr diagram. Start Alll from Bonni E Alll from Evan Outcom E (Nonattachd) E (Nonattachd) (Attachd) (Attachd) Thr ar four possibl alll pairs (outcoms). Two of ths pairs, and, rsult in attachd arlobs. Th probability that Bonni and Evan s baby 2 1 will hav attachd arlobs is or 4, 2. You can also find th probabilitis by making a Bonni tabl such as th on at th right. List Evan s E allls along th sid and Bonni s allls on top. Th four whit squars show th possibl Evan E combinations. E Bonni and Evan s chart is somtims calld a Punntt squar by gnticists. A Punntt squar is a chart which prdicts all possibl gn combinations. Punntt squars ar namd for an English gnticist, Rginald Punntt. H discovrd som basic principls of gntics. H studid th fathr color traits of chickns in ordr to quickly dtrmin whthr chickns wr mal or fmal whn thy wr born. For: Information about Punntt squars Wb Cod: am How Likly Is It?

5 Problm 4.2 Applying Thortical Probability In Qustions A C, xamin ach family situation and answr th qustions. A. Dasan s mothr is xpcting hr third child. His mothr and fathr both hav th arlob allls E. 1. What is th probability that Dasan s nw sibling will hav attachd arlobs? 2. What is th probability that his nw sibling will hav nonattachd arlobs? B. Goff s arlob allls ar EE and Mali s arlob allls ar E.What is th probability that thir child will hav nonattachd arlobs? C. Both of Eiln s parnts hav attachd arlobs. What is th probability that Eiln has attachd arlobs? Homwork starts on pag 62. Thr ar many othr traits you can study in th way you studid arlobs. For xampl, having crtain charactristics (dimpls, curly or wavy hair, and widow s pak) is dominant ovr not having th charactristics. Thr ar dominant traits that do not show up vry oftn in th population. For xampl, th trait for six fingrs on on hand is a dominant trait, and fiv fingrs is a rcssiv trait. Bcaus only a fw popl carry th alll for six fingrs, vry fw popl ar born with this trait. Grg Harris, a major-lagu basball playr from 1981 to 1995, has six fingrs on on hand. H had a spcially dsignd, rvrsibl six-fingrd glov. In 1995, h bcam th only pitchr sinc 1900 to pitch with both hands in a major-lagu gam. For: Information about gntic traits Wb Cod: am-9031 Invstigation 4 Probability, Gntics, and Gams 59

6 4.3 Rollr Drby Hav you vr figurd out a stratgy for winning a gam? Now that you know about making tabls and diagrams to find probabilitis, you can us ths tools to find winning stratgis for gams. In this problm, you play a two-tam gam calld Rollr Drby. Each tam nds a gam board with columns numbrd 1 12, a pair of numbr cubs, and 12 markrs (such as coins, buttons, or small blocks). Rollr Drby Ruls 1. Each tam placs its 12 markrs into thir columns in any way it chooss. 2. Each tam rolls a numbr cub. Th tam with th highst roll gos first. 3. Tams tak turns rolling th two numbr cubs. Thy rmov a markr from th column on thir board with th sam numbr as th total sum of th numbrs on th numbr cubs. If th column is mpty, th tam dos not gt to rmov a markr. 4. Th first tam to rmov all th markrs from its board wins. As you play, think about stratgis for winning and how probability rlats to your stratgis. 60 How Likly Is It?

7 Problm 4.3 Analyzing a Gam A. Play th gam at last twic. For ach gam, rcord th stratgis you us to plac your markrs on th board. Also, rcord how many tims ach sum is rolld. What is a good stratgy for placing your markrs on th gam board? B. 1. Which sums sm to occur most oftn? 2. Which sums do not com up vry oftn? C. Find all th possibl outcoms (numbr pairs) of rolling two numbr cubs. Find th sums for ach of ths outcoms. 1. Ar all th sums qually likly? Explain. 2. How many ways can you gt a sum of 2? 3. What is th probability of gtting a sum of 4? 4. What is th probability of gtting a sum of 6? 5. Which sums occur most oftn? D. Now that you hav lookd at th possibl outcoms of th Rollr Drby gam, do you hav any nw stratgis for winning? Explain. Homwork starts on pag 62. Galilo was an Italian physicist, astronomr, and mathmatician. H is famous for hlping dvlop a modl in which th sun was th cntr of th univrs. H also studid problms in probability similar to th ons you hav sn. A famous problm h workd on involvd rolling thr numbr cubs. H lookd at th possibilitis for gtting a sum of 9 or a sum of 10. A sum of 9 is mad using six groups of numbrs: (1, 2, 6), (1, 3, 5), (1, 4, 4), (2, 2, 5), (2, 3, 4), and (3, 3, 3). A sum of 10 is mad using six othr groups of numbrs: (1, 3, 6), (1, 4, 5), (2, 2, 6), (2, 3, 5), (2, 4, 4), and (3, 3, 4). What puzzld popl is that, whn thy did xprimnts, th sum of 10 occurrd mor oftn. By making a diagram similar to a counting tr, Galilo showd th thortical probability matchd th xprimntal rsults. Thr ar actually 25 combinations that hav a sum of 9 and 27 combinations that hav a sum of 10. For: Information about Galilo Wb Cod: am-9031 Invstigation 4 Probability, Gntics, and Gams 61