{"id":20994,"date":"2019-11-09T18:15:28","date_gmt":"2019-11-09T23:15:28","guid":{"rendered":"http:\/\/chcollins.com\/100Billion\/?p=20994"},"modified":"2023-10-14T12:43:27","modified_gmt":"2023-10-14T16:43:27","slug":"are-streaky-players-better","status":"publish","type":"post","link":"https:\/\/chcollins.com\/100Billion\/2019\/11\/are-streaky-players-better\/","title":{"rendered":"Are Streaky Players Better?"},"content":{"rendered":"

Asked & Answered 7.0<\/strong><\/em><\/h4>\n

You are the first-year coach of the Texas Lady Longhaulers of the WNBA.\u00a0 This morning finds you dispirited after an embarrassing 88-60 loss to the Nashville Wynettes last night.\u00a0 You need a more productive starting lineup for tomorrow’s game.\u00a0 Where to begin?<\/p>\n

Point guard is your most important position and your roster choices are Maya Thomas and Tamika DeShields.\u00a0 They are good but very different players.\u00a0 Maya is a steady shooter who has a 50% chance of sinking each shot.\u00a0 Tamika, on the other hand, is streaky: if she sinks a shot, she makes her next one 80% of the time; but if she misses, she tends to miss again, 80% of the time.<\/p>\n

So which point guard would you start, Steady Maya or Streaky Tamika?<\/p>\n

I thought I would make this edition of Asked and Answered<\/em> more interactive than usual and so I asked readers to weigh in with their answers.\u00a0 There were 7 responses, namely: Steady = 4 | Streaky =2 | No Difference = 0 | It Depends = 1.\u00a0 Who was right?<\/p>\n

First a little table-setting.\u00a0 Maya and Tamika are fictional stand-ins.\u00a0 Top players in the WNBA make 15-20 field-goal (two-point) attempts per game and sink 45-50% of those.\u00a0 Maya’s performance is realistic, but whether Tamika’s streakiness is seen in actual players is a question for sports statisticians to answer.<\/p>\n

Maya’s expected performance is easy to calculate.\u00a0 Assuming she makes 20 attempts and sinks 50% of them, you may expect 10 field goals from her in a typical game, give or take.\u00a0 The probability P(n)<\/em> of Maya scoring exactly\u00a0k<\/em> field goals in n <\/em>shots is given by<\/p>\n

P(n) = \\frac{n!}{k! (n-k)!} \\;p^{k} q^{n-k}<\/span><\/code><\/p>\n

where p<\/em> is the probability of making any given shot and q<\/em> is the probability of missing it.\u00a0 [This is the well-known formula for the binomial probability distribution<\/a>.\u00a0 Exclamation points denote the factorial operation<\/a> — they do not express my surprise.]\u00a0 So the chance that Maya will score 10 +\/- 1 field goals is about 50 percent:<\/p>\n\n\n\n\n\n\n
Chance of 9 scores<\/td>\n0.160<\/td>\n<\/tr>\n
Chance of 10 scores<\/td>\n0.176<\/td>\n<\/tr>\n
Chance of 11 scores<\/td>\n0.160<\/td>\n<\/tr>\n
Total<\/td>\n0.497<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Predicting what to expect from Tamika is more complicated.\u00a0 As I learned while studying this problem, her performance is an example of a Markov chain<\/em><\/a>.\u00a0 This is best explained by the diagram below.\u00a0 Tamika starts the game in the Initial<\/em> box.\u00a0 After taking her first shot, she moves either to the Scored<\/em> box (blue) or to the Missed<\/em> box (violet).\u00a0 The number next to each path shows the chance that she will follow that path when she shoots.<\/p>\n

\"\"<\/a><\/p>\n

Every time Tamika shoots, she moves along a path.\u00a0 Some paths return to the same box.\u00a0 For instance, when Tamika is in the Scored\u00a0<\/em>box (i.e., she sank her last shot), she has an 80% chance (0.8) of circling back to the Scored<\/em> box with her next shot.\u00a0 Otherwise she moves over to the Missed<\/em> box.\u00a0 And so on.<\/p>\n

No paths lead to the Initial<\/em> box because, when taking a shot, Tamika only scores or misses. In my intro, I failed to specify how Tamika typically performs on her first shot.\u00a0 For now, assume that Tamika starts the game cold as if she has just missed.\u00a0 This implies that her chance of moving from Initial<\/em> to Missed <\/em>is 80% and from Initial<\/em> to Scored<\/em> is 20%.<\/p>\n

The number of times Tamika visits the Scored<\/em> box gives us the number of field goals you may expect her to score during the game.\u00a0 But how do you calculate that?\u00a0 Luckily, thanks to folks like David L. Deever, professor emeritus of Otterbein University in Ohio, there are such things as Markov chain calculators<\/a>.\u00a0 You enter the path information into a table, and the calculator returns the probability that a given box is occupied on the n<\/em>th step.<\/p>\n

Here are the results Dr. Deever’s calculator produced for Tamika.\u00a0 The Scores<\/em> column shows the expected number of times Tamika has scored after taking n<\/em> shots:<\/p>\n

\"\"<\/a>The table shows that, after 20 shots, Tamika’s expected number of scores is 9.25, which is 0.75 less than the 10 scores you can expect from Maya.\u00a0 Tamika never fully recovers from her cold start, and she will trail Maya by 0.75 scores (on average) forever.<\/p>\n

So, if you had decided to start Steady Maya, your answer would be correct.<\/p>\n

But perhaps you assumed Tamika has a 50\/50 chance of scoring\/missing on her first shot, after which the 80\/20 rule would apply.\u00a0 If that were the case, then Tamika would not fall behind Maya at all.\u00a0 Each player would score 10 times in 20 shots, on average.\u00a0 So if you guessed there would be no difference, your answer would also be correct.<\/p>\n

\u2022 \u2022 \u2022\u00a0<\/p>\n

Tomorrow has turned into today, and your team is in the final minutes of its next game.\u00a0 Since Maya had a slight edge over Tamika, you decided to start her today.\u00a0 Unfortunately, your Lady Longhaulers have allowed too many turnovers, and they have fallen behind by 10 points.\u00a0 You figure you will get five more possessions before the final buzzer.\u00a0 To have any chance to win, your team will need to score on every one of those possessions.<\/p>\n

You call a time-out.\u00a0 Do you stay with Steady Maya or do you send in Streaky Tamika?<\/p>\n

This gets interesting.\u00a0 You need a player to sink five shots in a row.\u00a0 The chance that Maya can do this is 0.5 (the probability of her sinking any one shot) to the fifth power, or 3.1%. The chance that Tamika can do it, coming in cold, is 0.2 (her first-shot success rate) times 0.8 (her repeat success rate) to the fourth power, or 8.2%.\u00a0 Tamika is more than twice as likely to tie the score than Maya, though her chances are still slim.<\/p>\n

So if your answer was to send in Streaky Tamika, you would be correct.\u00a0 And this means that if your original answer was “it depends” then you would also be correct.\u00a0 Now, all Tamika has to do is sink her first shot.\u00a0 And the next.\u00a0 And the next…<\/p>\n

With that, your time-out is over.\u00a0 May you enjoy the final minutes of your coaching career.<\/p>\n

\u2022 \u2022 \u2022\u00a0<\/p>\n

David L. Deever, the author of journal articles<\/a> as well as the Markov calculator that I used,\u00a0 taught his last mathematics class<\/a> at Otterbein University in 2003, ending a 37-year career.\u00a0 His Facebook page (which has not been updated since 2013) reveals Dr. Deever to be a kind person, concerned citizen and a liberal in good standing.\u00a0 I thank Dr. Deever for his contributions and I wish him good health.<\/p>\n","protected":false},"excerpt":{"rendered":"

Asked & Answered 7.0 You are the first-year coach of the Texas Lady Longhaulers of the WNBA.\u00a0 This morning finds you dispirited after an embarrassing 88-60 loss to the Nashville Wynettes last night.\u00a0 You need a more productive starting lineup … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[58],"tags":[],"class_list":["post-20994","post","type-post","status-publish","format-standard","hentry","category-asked-and-answered"],"_links":{"self":[{"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/posts\/20994","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/comments?post=20994"}],"version-history":[{"count":92,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/posts\/20994\/revisions"}],"predecessor-version":[{"id":31325,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/posts\/20994\/revisions\/31325"}],"wp:attachment":[{"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/media?parent=20994"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/categories?post=20994"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/chcollins.com\/100Billion\/wp-json\/wp\/v2\/tags?post=20994"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}