{"id":1045,"date":"2015-09-13T17:04:20","date_gmt":"2015-09-13T09:04:20","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=1045"},"modified":"2024-02-16T21:09:14","modified_gmt":"2024-02-16T13:09:14","slug":"a-probability-puzzle-from-dilberts-author","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2015\/09\/a-probability-puzzle-from-dilberts-author\/","title":{"rendered":"A Probability Puzzle From Dilbert’s Author"},"content":{"rendered":"

I’ve been reading this book<\/a>, by Scott Adams, the author of Dilbert. Inside, I found a probability puzzle!<\/p>\n

Scott Adams talks about Volleyball games, and how he noticed that the team that reaches 17 first usually wins. (A win in volleyball is 25 points.)<\/p>\n

<\/p>\n

The probability puzzle is: what, exactly, is the chance of this happening? Scott Adams probably doesn’t realize he’s posed a probability puzzle, but he has!<\/p>\n

Here’s how I solved it. If you want to have a crack at it yourself, stop reading now, and come back later.<\/p>\n

I imagined that the outcome of a rally is decided, effectively, by random chance – team A wins p of the time, and team B wins 1-p of the time. For simplicity, I’ll write 1-p as q sometimes.<\/p>\n

First I worked out Pn,k<\/sub>, the chance that team B has k\u00a0points when team A scores their nth point.<\/p>\n

This is really easy to work out – the simplest method is to visit Wikipedia and read about the Negative Binomial distribution<\/a>. There you’ll find\u00a0a simple formula for\u00a0Pn,k<\/sub>, namely<\/p>\n

Pn,k<\/sub>\u00a0= n+k-1<\/sup>Ck<\/sub> pn<\/sup>qk<\/sup><\/p>\n

Next, I said\u00a0Qn,k<\/sub>\u00a0is the chance that team A gets n points before team B gets k. For example,\u00a0\u00a0Q25,25<\/sub>\u00a0would be the chance that team A wins the volleyball match.<\/p>\n

Qn,k<\/sub>\u00a0is the sum of\u00a0\u00a0Pn,i<\/sub>\u00a0as i\u00a0ranges from 0 to k-1. There may be a way to convert\u00a0this sum into a simple formula, but I was going to ask a computer to do the heavy lifting anyway, so I didn’t bother.<\/p>\n

Next, I said\u00a0\u00a0Rn,m<\/sub>\u00a0is the chance that team A gets m points first, and then gets to n points first.<\/p>\n

When team A gets to m points first, team B could have any number of points from 0 to m-1. The chance of each of these is given by the Pn,k<\/sub>. Then, for A to get to n points before B, they need another n-m points before B can clinch another n-k. Since m is bigger than k, you’d think this would be easy. Whatever it is, it’s given by\u00a0\u00a0Qn-m,n-k<\/sub><\/p>\n

To find\u00a0Rn,m<\/sub>\u00a0we can add the products \u00a0Pm,k<\/sub>\u00a0\u00a0\u00d7\u00a0Qn-m,n-k<\/sub>\u00a0as k ranges from 0 to m-1.<\/p>\n

R17,25<\/sub>\u00a0is not quite the answer to Scott Adams’ puzzle, because we don’t really care which team wins.<\/p>\n

Rn,m<\/sub>\u00a0depends on the chance of A winning, which is p. We can say\u00a0\u00a0Rn,m<\/sub>\u00a0=\u00a0\u00a0Rn,m<\/sub>(p). The answer to the puzzle is\u00a0Sn,m<\/sub>\u00a0=\u00a0\u00a0Rn,m<\/sub>(p) +\u00a0\u00a0Rn,m<\/sub>(1-p).<\/p>\n

Each formula given above is a relatively simple sum, although disentangling the sums into a single formula involving p might be quite difficult.<\/p>\n

It turns out that S17,25<\/sub>\u00a0is actually very high. The lowest it gets is 80.9%, when the teams are perfectly matched.<\/p>\n

Here’s a graph of \u00a0S17,25<\/sub>\u00a0or different values of p:<\/p>\n

\"Plot<\/a>
Plot of the probabilities for the Dilbert author’s probability puzzle<\/figcaption><\/figure>\n

As you can see, even if the teams are slightly mismatched, the probability of confirming the Dilbert author’s observation shoots up dramatically. It’s no wonder then.<\/p>\n

In fact, you can often pick the winner as soon as one of the teams reaches 13 points. Here’s the graph of \u00a0S13,25<\/sub><\/p>\n

\"The<\/a>
The chance that the first team to reach 13 points ends up winning<\/figcaption><\/figure>\n","protected":false},"excerpt":{"rendered":"

I’ve been reading this book, by Scott Adams, the author of Dilbert. Inside, I found a probability puzzle! Scott Adams talks about Volleyball games, and how he noticed that the team that reaches 17 first usually wins. (A win in volleyball is 25 points.)<\/p>\n","protected":false},"author":1,"featured_media":1046,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,258],"tags":[543,108,542],"_links":{"self":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/1045"}],"collection":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/comments?post=1045"}],"version-history":[{"count":2,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/1045\/revisions"}],"predecessor-version":[{"id":1274,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/1045\/revisions\/1274"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media\/1046"}],"wp:attachment":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media?parent=1045"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/categories?post=1045"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/tags?post=1045"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}