tag:theconversation.com,2011:/fr/topics/playing-cards-36159/articlesPlaying cards – The Conversation2018-01-17T12:24:13Ztag:theconversation.com,2011:article/901302018-01-17T12:24:13Z2018-01-17T12:24:13ZHow to avoid a sucker bet – with a little help from maths<figure><img src="https://images.theconversation.com/files/202007/original/file-20180116-53302-1lsj625.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A Friend in Need (1903).</span> <span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File%3AA_Friend_in_Need_1903_C.M.Coolidge.jpg">Cassius Marcellus Coolidge</a></span></figcaption></figure><p>Sitting in a bar, you start chatting to a man who issues you a challenge. He hands you five red and two black cards. After shuffling, you lay them on the bar, face down. He bets you that you cannot turn over three red cards. And to help you, he explains the odds.</p>
<p>When you draw the first card, the odds are 5-2 (five red cards, two black cards) in favour of picking a red card. The second draw is 4-2 (or 2-1) and the third draw is 3-2. Each time you draw a card the odds appear to be in your favour, in that you have more chance of drawing a red card than a black card. So, do you accept the bet?</p>
<p>If you answered yes, perhaps it’s time for you to go over your maths. It’s a foolish bet. The odds given above are only for a perfect draw. The real odds of you being able to carry out this feat are actually 5-2 against you. That is, for every seven times you play, you’ll lose five times.</p>
<h2>Odds against you</h2>
<p>This type of bet is often called a proposition bet, which is defined as a wager on something that seems like a good idea, but for which the odds are actually against you, often very much against you, perhaps even making it impossible for you to win.</p>
<p>Let’s assume that you took the bet and, almost inevitably, lost money. But this is just for fun, right? So your new “friend” suggests a way that you can get your money back. He takes two more red cards and hands them to you, so you now have seven red cards and two black cards. You shuffle the nine cards and lay them out, face down, in a three by three grid. He bets you even money that you can’t pick out a straight line (vertical, horizontal or diagonal) that has only red cards.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/201956/original/file-20180115-101495-69xi0s.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/201956/original/file-20180115-101495-69xi0s.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=840&fit=crop&dpr=1 600w, https://images.theconversation.com/files/201956/original/file-20180115-101495-69xi0s.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=840&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/201956/original/file-20180115-101495-69xi0s.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=840&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/201956/original/file-20180115-101495-69xi0s.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1055&fit=crop&dpr=1 754w, https://images.theconversation.com/files/201956/original/file-20180115-101495-69xi0s.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1055&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/201956/original/file-20180115-101495-69xi0s.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1055&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Nine Card Hustle.</span>
<span class="attribution"><span class="source">Graham Kendall created image</span></span>
</figcaption>
</figure>
<p>Intuitively, this might sound like a better bet and the odds are actually evens if the two black cards are next to each other in a corner (see image). In total there are eight lines to choose from and four contain only red cards, and four contain a black card. But that is as good as it gets.</p>
<p>If the black cards are in opposite corners then you can only win by choosing the centre horizontal or vertical row so the odds are 6-2 (or 3-1) against you winning. Every other layout gives you three winning lines and five losing lines. This bet only has 12 ways of succeeding, against 22 ways of you losing. Hardly an even-chance bet.</p>
<h2>Have another go</h2>
<p>Try to evaluate the odds for this proposition bet.</p>
<p>You shuffle a pack of cards and cut it into three piles. You are offered even money that one of the cards on top of the piles will be a picture card (a jack, queen or king). That is, if a picture card shows up, you lose. Do you think this is a good bet?</p>
<p>One way of reasoning is that there are only 12 losing cards against 40 winning cards, so the odds look better than evens? But this is the wrong way of looking at it. It is really what’s known as a <a href="https://mathigon.org/world/Combinatorics">combinatorics</a> problem. We should also realise that we are just choosing three cards at random.</p>
<p>There are 22,100 ways of choosing three cards from a 52 card deck. Of these, 12,220 will contain at least one picture card – so you lose – meaning that 9,880 will not contain a picture card – when you win. If you translate this to odds, you will lose fives times out of every nine times you play (5-4 against you). The even chance bet you have been offered is not the good value that you thought it was and you will lose money if you play a few times.</p>
<h2>A Final Example</h2>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/202011/original/file-20180116-53324-1w4i08x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/202011/original/file-20180116-53324-1w4i08x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=759&fit=crop&dpr=1 600w, https://images.theconversation.com/files/202011/original/file-20180116-53324-1w4i08x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=759&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/202011/original/file-20180116-53324-1w4i08x.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=759&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/202011/original/file-20180116-53324-1w4i08x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=954&fit=crop&dpr=1 754w, https://images.theconversation.com/files/202011/original/file-20180116-53324-1w4i08x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=954&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/202011/original/file-20180116-53324-1w4i08x.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=954&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Coin Toss.</span>
<span class="attribution"><span class="source">ICMA Photos</span></span>
</figcaption>
</figure>
<p>We can all agree that you have a 50/50 chance of guessing heads or tails in a coin toss. But if you toss the coin ten times, would you expect to see five heads and five tails? If you were offered odds of 2-1 to try this, would you take the bet? You’d be a sucker if you did.</p>
<p>Five heads and five tails will occur more often than any other combination, but there are many other ways that ten flips of a coin can land. In fact, the bet is 5-2 against you.</p>
<p>Another name for a proposition bet is the “sucker” bet, and there is no surprise who the sucker is. But don’t feel too bad. We are all generally very poor at evaluating true odds. A famous example is the <a href="https://theconversation.com/the-monty-hall-problem-going-with-your-gut-will-get-your-goat-14195">Monty Hall Problem</a>. Even mathematicians could not agree on the right answer to this seemingly simple problem.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/4Lb-6rxZxx0?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Monty Hall Problem - Numberphile.</span></figcaption>
</figure>
<p>We have focused on bets where it is difficult, especially when under the pressure of deciding whether to bet or not, to calculate the true odds. But there are many <a href="https://www.youtube.com/user/propbetonly">other proposition bets</a> that do not rely on calculating odds. And there are many other sucker bets, with probably the most famous being the Three Card Monty.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/YnXUe3wV-4M?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Three Card Monty.</span></figcaption>
</figure>
<p>If faced with this type of bet, what is the best thing you can do? I’d suggest you simply walk away.</p><img src="https://counter.theconversation.com/content/90130/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Graham Kendall does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>With a proposition bet, the odds are always against you.Graham Kendall, Professor of Computer Science and Provost/CEO/PVC, University of NottinghamLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/739332017-03-03T10:10:34Z2017-03-03T10:10:34ZThe Conversation weekly quiz – #1<figure><img src="https://images.theconversation.com/files/159160/original/image-20170302-14717-3l83hz.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">via shutterstock.com</span></span></figcaption></figure><p><strong>At The Conversation, we’re constantly finding out amazing new facts about the world from the academics who write for us. So to test how closely you’ve been reading this week, we’ll be running a short quiz each Friday. Answers below … if you need them.</strong> </p>
<p>1: Which 16-year-old future British monarch went to Japan and got famous Japanese tattoo artist Hori Chiyo to ink him?</p>
<p>2: Researchers have found that 55% of people involved in which activity on their mobile phones were under the age of 16? </p>
<p>3: Which iconic funk drummer, who played for Otis Redding and James Brown – and whose famous Funky Drummer drum break has been sampled in over 1,300 songs – died in late February? </p>
<p>4: Which is the UK’s third smallest city? </p>
<p>5: How much does the British curry industry contribute to the UK economy?</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/159155/original/image-20170302-14717-urc5tv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/159155/original/image-20170302-14717-urc5tv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=383&fit=crop&dpr=1 600w, https://images.theconversation.com/files/159155/original/image-20170302-14717-urc5tv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=383&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/159155/original/image-20170302-14717-urc5tv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=383&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/159155/original/image-20170302-14717-urc5tv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=481&fit=crop&dpr=1 754w, https://images.theconversation.com/files/159155/original/image-20170302-14717-urc5tv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=481&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/159155/original/image-20170302-14717-urc5tv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=481&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">British cuisine.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/jryde/4336201294/sizes/o/">J</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>6: What function do xylem elements perform inside plants? </p>
<p>7: Which country in Eastern Europe sold people to West Germany and Israel during the Cold War? </p>
<p>8: The 19th-century scientist Luigi Galvani used electricity to animate the dismembered limbs of which animals? </p>
<p>9: Roughly how old are the world’s earliest known fossils, which have recently been discovered in Canada?</p>
<p>10: How many times do mathematicians recommend you shuffle a pack of cards to give a fair deck?</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/159154/original/image-20170302-14717-2rv4j0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/159154/original/image-20170302-14717-2rv4j0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=397&fit=crop&dpr=1 600w, https://images.theconversation.com/files/159154/original/image-20170302-14717-2rv4j0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=397&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/159154/original/image-20170302-14717-2rv4j0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=397&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/159154/original/image-20170302-14717-2rv4j0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=499&fit=crop&dpr=1 754w, https://images.theconversation.com/files/159154/original/image-20170302-14717-2rv4j0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=499&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/159154/original/image-20170302-14717-2rv4j0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=499&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">One last time.</span>
<span class="attribution"><span class="source">Daniil Yanopulo via www.shutterstock.com</span></span>
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</figure>
<h2>Answers</h2>
<ol>
<li><p><a href="https://theconversation.com/ink-stigma-the-japanese-tattoo-artists-fighting-back-72943">King George V</a>. </p></li>
<li><p><a href="https://theconversation.com/why-teaching-children-about-porn-and-sexting-is-a-step-in-the-right-direction-73001">Sexting</a>. </p></li>
<li><p><a href="https://theconversation.com/the-story-of-the-funky-drummer-the-most-exploited-man-in-modern-music-73473">Clyde Stubblefield</a>. </p></li>
<li><p>The <a href="https://theconversation.com/small-cities-can-offer-just-as-much-culture-as-larger-ones-73218">City of London</a>.</p></li>
<li><p>More than <a href="https://theconversation.com/tough-immigration-laws-are-hitting-britains-curry-houses-hard-72942">£4 billion</a> a year. </p></li>
<li><p>They <a href="https://theconversation.com/scientists-create-electric-circuits-inside-plants-73711">carry water</a> from the roots to the leaves. </p></li>
<li><p><a href="https://theconversation.com/people-have-been-used-as-bargaining-chips-before-by-romanias-nicolae-ceau-escu-73141">Romania</a>. </p></li>
<li><p><a href="https://theconversation.com/awesome-erotic-everyday-the-literary-story-of-electricity-73624">Frogs</a>. </p></li>
<li><p><a href="https://theconversation.com/how-we-discovered-the-worlds-oldest-fossils-73802">Between 3.8 billion and 4.3 billion years old</a>. </p></li>
<li><p><a href="https://theconversation.com/heres-the-best-way-to-shuffle-a-pack-of-cards-with-a-little-help-from-some-maths-73176">Seven</a> (or 11, depending on how you measure fairness).</p></li>
</ol><img src="https://counter.theconversation.com/content/73933/count.gif" alt="The Conversation" width="1" height="1" />
Test your knowledge in our first ever weekly quiz.Gemma Ware, Head of AudioLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/731762017-02-22T16:10:22Z2017-02-22T16:10:22ZHere’s the best way to shuffle a pack of cards – with a little help from some maths<figure><img src="https://images.theconversation.com/files/157873/original/image-20170222-1350-i22us5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>Shuffling a pack of cards isn’t as easy as you think, not if you want to truly randomise the cards. Most people will give a pack a few shuffles with the overhand or riffle methods (where the pack is split and the two halves are interweaved). But research has shown this isn’t enough to produce a sufficiently random order to make sure the card game being played is completely fair and to prevent people cheating.</p>
<p>As I wrote <a href="https://theconversation.com/how-to-beat-the-casino-legally-58944">in a recent article</a> about card counting, not having an effective shuffling mechanism can be a serious problem for casinos:</p>
<blockquote>
<p>Players have used shuffle tracking, where blocks of cards are tracked so that you have some idea when they will appear. If you are given the option to cut the pack, you try and cut the pack near where you think the block of cards you are tracking is so that you can bet accordingly. A variant on this is to track aces as, if you know when one is likely to appear, you have a distinct advantage over the casino.</p>
</blockquote>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/uRH9njr0IgE?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Card Counting and Shuffle Tracking in Blackjack.</span></figcaption>
</figure>
<p>So how can you make sure your cards are well and truly shuffled?</p>
<p>To work out how many ways there are of arranging a standard 52-card deck, we multiply 52 by all the numbers that come before it (52 x 51 x 50 … 3 x 2 x 1). This is referred to as “52 factorial” and is usually written as “52!” by mathematicians. The answer is so big it’s easier to write it using scientific notation as 8.0658175e+67, which means it’s a number beginning with 8, followed by 67 more digits.</p>
<p>To put this into some sort of context, if you dealt one million hands of cards every second, it would take you 20 sexdecillion, or 20,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, years to deal the same number of hands as there are ways to arrange a deck of cards.</p>
<p>You would think that it would be easy to get a random order from that many permutations. In fact, every arrangement is, in a sense, random. Even one where the cards are ordered by suit and then rank could be considered random. It is only the interpretation we put on this order that would make most people not consider it random. This is the same as the idea that the lottery is less likely to throw up the numbers one to six, whereas in reality this combination is just as probable as any other.</p>
<p>In theory, you could shuffle a deck so that the cards emerged in number order (all the aces, followed by all the twos, followed by all the threes and so on), with each set of numbers in the same suit order (say spades, hearts, diamonds and clubs). Most people would not consider this random, but it is just as likely to appear as any other specific arrangement of cards (very unlikely). This is an extreme example but you could come up with an arrangement that would be seen as random when playing bridge because it offered the players no advantage, but wouldn’t be random for poker because it produced consistently strong hands.</p>
<p>But what would a casino consider random? Mathematicians have developed several ways of measuring how random something is. Variation distance and separation distance are two measures calculated by mathematical formulas. They have a value of 1 for a deck of cards in perfect order (sorted by numbers and suits) and lower values for more mixed arrangements. When the values are less than 0.5, the deck is considered randomly shuffled. More simply, if you can guess too many cards in a shuffled deck, then the deck is not well shuffled.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/AxJubaijQbI?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">The Best (and Worst) Ways to Shuffle Cards - Numberphile.</span></figcaption>
</figure>
<p>Persi Diaconis is a mathematician who has been studying card shuffling for over 25 years. Together with and Dave Bayer, <a href="http://projecteuclid.org/euclid.aoap/1177005705">he worked out that</a> to produce a mathematically random pack, you need to <a href="http://dx.doi.org/10.1214/009117905000000675">use a riffle shuffle</a> seven times if you’re using the variation distance measure, or 11 times using the separation distance. The overhand shuffle, by comparison, requires 10,000 shuffles to achieve randomness.</p>
<p>“The usual shuffling produces a card order that is far from random,” Diaconis <a href="http://www.nytimes.com/1990/01/09/science/in-shuffling-cards-7-is-winning-number.html">has said</a>. “Most people shuffle cards three or four times. Five times is considered excessive”.</p>
<p>But five is still lower than the number required for an effective shuffle. Even dealers in casinos rarely shuffle the required seven times. The situation is worse when more than one deck is used, as is the case in blackjack. If you are shuffling two decks, you should <a href="http://www.nytimes.com/1990/01/09/science/in-shuffling-cards-7-is-winning-number.html">shuffle nine times</a> and for six decks you need to shuffle twelve times.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/Uf-AuZUP-o0?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Shuffle like a casino dealer.</span></figcaption>
</figure>
<p>Many casinos now use <a href="http://vitalvegas.com/a-rare-look-inside-a-casino-automatic-card-shuffler/">automatic shuffling machines</a>. This not only speeds up the games but also means that shuffles can be more random, as the machines can shuffle for longer than the dealers. These shuffling machines also stop issues such as <a href="https://theconversation.com/how-to-beat-the-casino-legally-58944">card counting and card tracking</a>.</p>
<p>But even these machines are not enough. <a href="http://dx.doi.org/10.1214/12-AAP884">In another study</a>, Diaconis and his colleagues were asked by a casino to look at a new design of a card shuffling machine that the casino had built. The researchers found that the machine was not sufficiently random, as they simply did not shuffle enough times. But using the machine twice would resolve the problem.</p>
<p>So next time you’re at a casino, take a look at how many times the dealers shuffle. The cards may not be as random as you think they are, which could be to your advantage.</p><img src="https://counter.theconversation.com/content/73176/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Graham Kendall does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>When is a pack of cards truly random?Graham Kendall, Professor of Computer Science and Provost/CEO/PVC, University of NottinghamLicensed as Creative Commons – attribution, no derivatives.