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Calculating probabilityA simple diagram like the one below can be used to calculate the probability of anyoutcome. The sum of the probabilities of all possible outcomes for a random eventmust always add to 1. For a coin flip, there are two possible outcomes, either heads ortails. So if the coin is unbiased, and the coin flipping technique is also unbiased, thenthe probabilities are exactly half and half (½ or 0.5 or 50%).So, using the diagram, there are four possible outcomes on the second flip. Theprobability of getting HH on the second flip is ¼ (or 0.25). The probability of HT, TH,or TT are all also ¼. The probability of HHH on the third flip is 1/8 (or 0.125) and HHHHon the fourth is 1/16 (or 0.0625).H1st FlipHH2nd Flip3rd Flip4th TTTTTTTTTHTTTTTHTTHTTHHTTHTTHHTHHHTHHTTHTTHTHTHTTA faster way to calculate the probability of an outcome after any number of flips is tomultiply the probabilities of all the outcomes together. So the probability of HH is ½ ½ ¼. The probability of HHH is ½ ½ ½ 1/8 and of HHHH is ½ ½ ½ ½ 1/16.Calculating the probabilities and expected frequencies of clusters is beyond the scopeof primary school mathematics but students should be able to grasp the concept oftheir expected frequency and notice that their combined results match thesepredictions almost perfectly.The expected frequencies for each cluster size are provided in Table 5 in the StudentWorksheet. Each frequency in the table represents the average number of times youwould expect to see these clusters in a large number of sets of results from onehundred coin flips.Lesson Plan 19 – Flipping Coins Page 72008 ABC Science Online

Expected resultsThe table below contains ten sets of real results. The averages calculated in the table areplotted in pink on the graph JaneSuzie2HeadsBradClusterSizeMy 25ObservedAverage Frequencyfrequency201510512345678910Cluster Size(Number of consecutive heads and consecutive tails)Lesson Plan 19 – Flipping Coins Page 82008 ABC Science Online

Probability in the real worldThe Gambler’s FallacyThe Gambler‟s Fallacy is explained in the student worksheet. It arises when we know theprevious results of a series of random events like coin flips. Because we expect the 50/50 ruleto appear sooner or later, we incorrectly assume that the probability of getting either heads ortails changes over time. After a long series of heads, many people assume that the probabilityof tails on the next flip is higher than ½. In reality, the probability of tails is still exactly ½ everytime we flip the coin, no matter how many heads preceded it.This fallacy can obviously get gamblers into very serious trouble. To make matters worse, theodds of casino games are stacked slightly against the gambler (eg the inclusion of a green zeroon roulette wheels stacks the odds of black/red, or odd/even slightly in favour of the casino –even though every result is independent of the last and truly random, the green zeroguarantees the casino will take more winnings than the gambler given sufficient time).A second problem in addition to the Gambler‟s Fallacy is the tendency to remember large winsand to forget small losses even though those small losses may add up to be significantly higherin value than the occasional large wins.“Littlewood’s Law of Miracles”Renowned University of Cambridge mathematician John Littlewood once famously calculatedthe expected natural frequency of strange coincidences. He defined a 'miracle' as being a onein a million event we might believe to be so unlikely that we deem it to have 'real significance'.He assumes we hear, see and think things at a rate of roughly one per second. In an eight hourday, we experience around 30,000 such 'events' (60 seconds 60 minutes 8 hours 28,800). At this rate, we experience around one million 'events' per 30 day month (864,000 tobe precise). Most of these events are so mundane or expected that pass by unnoticed. Butevery now and then we experience some strange coincidence eg you‟re thinking about an oldfriend seconds before they call, or a song you‟re humming comes on the radio.A series of fortuitous events might make you feel especially lucky eg ten traffic lights turngreen as you approach, each giving you a clear run to an important appointment for which youare running late, followed by a parking space 'magically' opening up just as you arrive. Lessfortunate ones could make you feel 'cursed', such as ten red lights, no parking spaces withinten blocks for an appointment, followed by a toothache and an expensive parking ticket.If we agree that events with probabilities around one in a million could be called 'miracles', thenwe could expect one such 'miracle' around once per month. While 'Littlewood‟s Law of Miracles'shines some light on their statistical probability, it shouldn‟t diminish our enjoyment of thosepleasant, quirky coincidences. It could however, help anyone who endures three or moreunfortunate events in succession to stay positive and motivated, rather than sink into milddepression or suffer unwarranted anxieties.Lesson Plan 19 – Flipping Coins Page 92008 ABC Science Online

Activity 1 - HomeworkTable 1 – One Hundred Coin FlipsNameTossing a coin always results in either heads ortails but never both. No surprises there! Butsomething very strange lurks beyond thissimple fact. To find it, you’d need to flip a cointhousands of times. That would take ages butby contributing just one hundred results each,your class can discover this quirky fact ofnature 2621363Step 11464Flip a coin. Do not record your result.156516661767186819692070under TAILS in step 2.21712272Step 3484358536863787388839894090419142924393If the result is heads, follow the instructionsunder HEADS in Step 2.If the result is tails, follow the instructionsComplete the table on this page by flipping acoin one hundred times. For each toss, place atick in the H column if you flipped heads or theT column if you flipped tails.Please do not make any marks in the lastcolumn. You will use the A column for analysisin the next class activity.TAILSComplete the table on this page by pretendingto flip a coin one hundred times. There is no'right' or 'wrong' way to do this. Just record theresult you would expect each time.For each imaginary toss, mime the action oftossing and catching the coin. Then place a tickin the H column or in the T column.4494Please do not make any marks in the last4595column. You will use the A column for analysis4696in the next class activity.479748982008 ABC ScienceOnline499950100HTA

Activity 2 – AnalysisHow did your teacher know whose results were real and whose werefaked? As you discovered, it was actually very easy! Flipping a coinrepeatedly usually results in surprisingly long sets of heads or tails.These consecutive results are called clusters and they are much morecommon than we expect. In this activity, you will count the number ofdifferent sized clusters in your results.Table 2 – TotalsHeadsTailsStep 1 – Count the totalsFirst, count the total number of Heads and Tails you flipped. Record your#Hresults in Table 2.1 2Step 2 – Circle the clusters3Draw circles around consecutive groups of ticks (clusters) in Table 1 as568Count the number of ticks in each cluster and write this number in9column A. For example, write the number 1 in column A for single ticks.10Write the number 2 in column A for two consecutive ticks. Write the11number 3 in column A for clusters of three ticks, and so on.Count the number of single ticks you circled and write in column A.16Record this total row 1 of the Frequency column in Table 4. Next, countthe number of circles containing two ticks in column A. Record this15171819number in row 2 of the Frequency Column. Continue counting thenumber of circles (clusters) containing 3, 4, 5 and up to 10 ticks until youhave completed Table 2.2112 1314 42112Circle and count thecrosses in each clusterlike thisStep 5 – Check your resultsTable 3– Cluster FrequencyFrequency 12Step 4 – Count the clustersClusterSize 7Step 3 – Count the ticks in each clusterA3 4illustrated. Be sure to group consecutive results for flip #50 and flip #51.TCheck columnThis step will help you check that youcounted all the ticks and tallied all the1 1 clusters correctly.2 2 For each row, multiply the number in3 3 indicated in the „Check‟ column.4 4 5 5 6 6 7 7 clusters.8 8 If your „Check total‟ is not 100, you9 9 10Check total 10 the „Frequency‟ column by the numberFinally, add all the numbers in thecheck column and write the result nextto „Check total‟. If your result is 100,you have correctly counted all themade a mistake in Step 3 but don‟tworry! Just go back to Step 3 and tallythe clusters again until you‟ve foundthe problem.2008 ABC Science Online

Activity 3 – Combine ResultsNow that you have counted the clusters you recorded, it’s time to share your results. Thiswill increase your sample from one hundred results to one thousand results. Yourcombined results will reveal just how common those unexpectedly long clusters really are.Step 1 – Copy your resultsTranscribe your results from Tables 2 and 3 into the column labelled 'My Results' in Table 4 below.Step 2 – Share resultsShare your results with nine other students. Write each person‟s name at the top of the remainingcolumns in Table 4. Then copy their totals from Table 2 into the first two rows, and their results fromTable 3 in the rows labelled 1 to 10.Step 3 – Average resultsAdd all the numbers in the Total Heads row and write the answer in your science journal. To calculatethe average number of heads observed in all 1000 results, simply divide this number by 10. Recordthe result in the average column. Repeat for the Total Heads row.Next, calculate the average frequency of each cluster size observed in your set of data. Again, simplyadd all ten numbers in each row and divide the total by 10. Write these results in the Average column.Table 4 – Combining 1000 ResultsTotalHeadsTotalTails123456789102008 ABC Science Online109876543Average2ClusterSize1 My ResultsFrequency

Activity 4 – Surprisingly PredictableThe probability of getting heads or tails on a fair toss are exactly equal. So we can write theprobability of heads or tails as ½ (or 0.5). The probability of getting two heads in a row is ½ ½ ¼ (or 0.25). The probability of getting three heads in a row is ½ ½ ½ 1/8 and so on. Usingthese rules, mathematicians can calculate the expected frequency of any series of consecutiveresults. The expected frequencies results for 100 coin flips are provided in Table 5 below.Table 5 – Expected Cluster Frequencies for 100 Coin FlipsCluster 60.780.390.190.100.05This table tells you that if you flip a coin 100 times, you should expect to see 25 single (non-consecutive)results. You should also expect 12.5 pairs of consecutive heads or tails, 6.25 triples etc.Remember that real clusters are counted in whole numbers (integers) but their average frequency can be afraction.Step 1 – Comparing your resultsPlot your results from the Average column in Table 3 on the graph below. Use the blank space to theright of each grey bar for each cluster size.How closely did the average results of your combined results (representing 1000 coin flips) match theexpected average frequencies of each cluster size?Do you think the results would match more closely if you combined even more results?

Random doesn’t look randomIn this activity, you discovered that random results don‟t really look like random results to theuntrained eye. Most people are surprised to see five or six consecutive heads or tails in a row but inreality, these clusters are actually quite common. Our intuition about independent, random eventsdoes not match reality very well.Everybody knows the probability of getting heads or tails when you flip a coin are exactly equal. So, ina series of one hundred coin flips, we would expect to see fifty heads and fifty tails. This is whatpeople mean when they say fifty, fifty.For one hundred coin flips, fifty/fifty is roughly what you get but we‟re not too surprised by resultslike 42/58. The problem is that we expect the 50/50 rule to apply on much shorter time scales than itactually does.For ten coin flips, most people expect around five heads and five tails. But you know now that resultslike seven heads and three tails are not unusual. Such results wouldn‟t bother you much unless therewas something riding on the outcome. When there is something valuable at stake however, we caneasily fall into a trap called “The Gambler‟s Fallacy”.The Gambler’s Fallacyfallacynoun, plural fallacies.1. a deceptive, misleading, or false notion, belief, etc.: a popular fallacy.2. a misleading or unsound argument.3. deceptive, misleading, or false nature.4. Logic any of various types of erroneous reasoning that render arguments logically unsound.5. Obsolete deception.(From Macquarie Dictionary)The Gambler‟s Fallacy is an incorrect belief about the probability of independent random events suchas coin flips. We all know the probability of heads is always exactly half, every time you flip a fair coin.The problem arises after a series of three or more heads because we expect the fifty/fifty rule toappear. As a result, we assume that the probability of tails on the next flip must be greater thanfifty/fifty. In reality, the probability of tails is still exactly the same no matter how many headspreceded a flip. Even after ten consecutive heads, the probability of tails on the next flip is still fiftyper cent!Over hundreds of consecutive flips, the fifty/fifty rule holds true but you can never predict theoutcome of an individual flip.A coin toss doesn‟t mean very much unless we have something riding on the outcome. When there issomething valuable at stake however, we can easily become very sensitive about the result. If severalrandom events go the wrong way, we might start feeling „unlucky‟ or become depressed. In somesituations, this can even make us start feeling suspicious or acting completely out of character. To see howeasily this could happen, imagine how you would feel in the following situation:

A Random StoryOne day, you invite a new friend home from school. You walk and chat and stop for a rest inthe park on the way. As you turn the last corner into your street, you both spot something small andshiny lying on the footpath right in front of your house. You race to see what it could be lying thereon the grass. You arrive together to discover that the shiny object is a very expensive looking sportswatch. It has five separate dials controlling all kinds of high-tech functions including GPS, a depthgauge, timers, alarms and even a heart monitor! The shiny stainless steel wrist band is made ofhundreds of intricate, inter-connecting links. It‟s by far the most amazing watch you have ever seen.You pick it up and hurry to your room, close the door and start investigating how all thebuttons work together. The watch beeps and the display changes. The light button illuminates thedial in a cool blue-green glow. Your mother brings you a sandwich and a glass of milk and asks tosee what you‟ve got. She takes a long look at the watch and realising how precious it, decides youshould take it to the police. She drives you and your friend to the local station straight away.The police officer asks lots of questions. Where did you find it? At what time? Was thereanything else nearby? She completes an official form, records both your names and commends youfor your honesty. Then she offers you both a little ray of hope. Judging by the brand, she says shewouldn‟t be surprised if the owner gave you a reward for returning the watch.After three months, the officer calls and invites you to the station with your parents. Nobodyhas collected the watch! To your surprise, your friend is there too. The officer asks who will besigning for the watch. You look at each other and shrug.“Well, I saw it first”, your friend says.“No you didn‟t, we found it together!”, you reply. “And anyway, we found it at my house”, youhasten to add. But the policewoman says where you found it makes no difference at all. Seeing yourdilemma, she suggests you should toss a coin to decide who will keep the watch. It‟s the only fairway to decide. Your heart starts to race. Your eyes meet your friend‟s and suddenly, you don‟t likesuch good friends anymore. Your friend‟s dad takes a coin out from his pocket.“Here, best of three”, he says, handing you the coin. Your flip flies high in the air, the coin is ablur.“Heads”, your friend calls. Blast! It‟s heads. You flip again.“Heads!”, your friend calls again. The coin seems to hover forever. You‟re already one downand it‟s the best watch you‟ve ever seen. It just has to be tails this time! You slap the coin on yourwrist and slowly lift your hand but you can‟t believe your eyes. Heads again!“Well that settles it, I guess?”, the policewoman says. Your face turns bright red and everyonecan tell how disappointed you are. But while your friend‟s dad is signing the release form, youdecide to toss the coin again out of sheer frustration. Heads again. That‟s odd, you think so youtoss it again and this time, you really can‟t believe your eyes. Heads again! That‟s four in a row. Youflip it once more to be sure. Heads again!!!“Hey!”, you exclaim. “This coin is rigged!” Everyone turns to look at you.“I just flipped this coin three more times. That‟s five heads in a row. I want a rematch. Youcheated!”Your friend‟s father asks for the coin and flips it. Tails!“You‟re cheating! I want a new coin”, you protest. The police officer asks for the coin and flipsit three times but says nothing. She just gives you a look that makes you sink. It‟s half pity, halfdisappointment. Your mum has exactly the same expression on her face.“But it‟s true. I got five heads in a row! That‟s impossible. Try it again!” This time, the policeofficer just gives you a sad looking smile and hands the coin back to your friend‟s father. He slipsthe coin back in his pocket and you watch it disappear. Now you‟ll never be able prove the coin wasa fake or was it?2008 ABC Science Online

2008 ABC Science Online Lesson Plan 19 Flipping Coins Brief description Flipping a coin one hundred