{"id":1154,"date":"2023-12-19T03:33:57","date_gmt":"2023-12-19T03:33:57","guid":{"rendered":"https:\/\/www.allendowney.com\/blog\/?p=1154"},"modified":"2023-12-26T21:37:16","modified_gmt":"2023-12-26T21:37:16","slug":"what-are-the-odds","status":"publish","type":"post","link":"https:\/\/www.allendowney.com\/blog\/2023\/12\/19\/what-are-the-odds\/","title":{"rendered":"What are the odds?"},"content":{"rendered":"\n

Whenever something unlikely happens, it is tempting to ask, “What are the odds?”<\/p>\n\n\n\n

In some very limited cases, we can answer that question. For example, if someone deals you five cards from a well-shuffled deck, and you want to know the odds of getting a royal flush, we can answer that question precisely<\/a>. At least, we can if you are clearly referring to just this one hand.<\/p>\n\n\n\n

But if you’ve been playing poker regularly for a decade and then one night you are dealt a royal flush, it might not be clear, when you ask the question, whether you mean the odds of getting a royal flush on one deal, or one evening of play, or some time in your career, or once in all of the poker hands that have every been dealt. Those are different questions with very different answers — in fact, the first is close to 0 and the last is close to 1 (and known to be 1 in this universe).<\/p>\n\n\n\n

So, even in a highly constrained environment like a poker game, answering questions like this can be tricky. It’s even worse in real life. Say you go to college in Massachusetts and then two years later you visit Paris, go for a walk in the Tuileries Garden, and run into a friend from college. What are the odds? Now we have to define both “in how many attempts?” and “odds of what?” Meeting this friend in this particular place? Or any old friend in any unexpected place?<\/p>\n\n\n\n

Now let’s put all of this thinking to the test with an example, which is the most surprising thing that has happened to me since the time I ran into a college friend in Paris. Two days ago I was working on a heat vent in my house and wanted to attach this socket<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

to this screwdriver<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

But the socket takes a 1\/4 inch square<\/em> drive, and the screwdriver takes 1\/4 inch hex<\/em> bits. I figured there was probably an adapter that could connect them, but I didn’t have one. I thought about getting one, but then I found another way to do the job.<\/p>\n\n\n\n

Two days later I went for a walk and about 30 yards from my house, in the middle of the street, I saw a small bit of metal that I picked up just to get it out of the way. And when I looked more closely at what is was — it was a 1\/4 inch hex to 1\/4 inch square drive adapter.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

And here’s how it works.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

So, what are the odds of that? I don’t know, but if you have a non-zero prior for the existence of a benevolent deity, you might want to update it.<\/p>\n","protected":false},"excerpt":{"rendered":"

Whenever something unlikely happens, it is tempting to ask, “What are the odds?” In some very limited cases, we can answer that question. For example, if someone deals you five cards from a well-shuffled deck, and you want to know the odds of getting a royal flush, we can answer that question precisely. At least, we can if you are clearly referring to just this one hand. But if you’ve been playing poker regularly for a decade and then one…<\/p>\n

Read More Read More<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[1],"tags":[],"class_list":["post-1154","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"yoast_head":"\nWhat are the odds? - Probably Overthinking It<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.allendowney.com\/blog\/2023\/12\/19\/what-are-the-odds\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What are the odds? - Probably Overthinking It\" \/>\n<meta property=\"og:description\" content=\"Whenever something unlikely happens, it is tempting to ask, “What are the odds?” In some very limited cases, we can answer that question. 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If I have a tumor that I\u2019ve been told has a malignancy rate of 2% per year, does that compound? So after 5 years there\u2019s a 10% chance it will turn malignant? This turns out to be an interesting question, because the\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":1145,"url":"https:\/\/www.allendowney.com\/blog\/2023\/12\/12\/smoking-causes-cancer-2\/","url_meta":{"origin":1154,"position":1},"title":"Smoking Causes Cancer","author":"AllenDowney","date":"December 12, 2023","format":false,"excerpt":"In the preface of Probably Overthinking It, I wrote: Sometimes interpreting data is easy. For example, one of the reasons we know that smoking causes lung cancer is that when only 20% of the population smoked, 80% of people with lung cancer were smokers. If you are a doctor who\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2023\/12\/image-1.png?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2023\/12\/image-1.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2023\/12\/image-1.png?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2023\/12\/image-1.png?resize=700%2C400&ssl=1 2x"},"classes":[]},{"id":442,"url":"https:\/\/www.allendowney.com\/blog\/2020\/04\/13\/bayesian-hypothesis-testing\/","url_meta":{"origin":1154,"position":2},"title":"Bayesian hypothesis testing","author":"AllenDowney","date":"April 13, 2020","format":false,"excerpt":"I have mixed feelings about Bayesian hypothesis testing. On the positive side, it's better than null-hypothesis significance testing (NHST). And it is probably necessary as an onboarding tool: Hypothesis testing is one of the first things future Bayesians ask about; we need to have an answer. On the negative side,\u2026","rel":"","context":"In \"bayesian statistics\"","block_context":{"text":"bayesian statistics","link":"https:\/\/www.allendowney.com\/blog\/tag\/bayesian-statistics\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2020\/04\/image.png?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":353,"url":"https:\/\/www.allendowney.com\/blog\/2019\/12\/08\/handicapping-pub-trivia\/","url_meta":{"origin":1154,"position":3},"title":"Handicapping pub trivia","author":"AllenDowney","date":"December 8, 2019","format":false,"excerpt":"Introduction The following question was posted recently on Reddit's statistics forum: If there is a quiz of x questions with varying results between teams of different sizes, how could you logically handicap the larger teams to bring some sort of equivalence in performance measure?[Suppose there are] 25 questions and a\u2026","rel":"","context":"In \"binomial\"","block_context":{"text":"binomial","link":"https:\/\/www.allendowney.com\/blog\/tag\/binomial\/"},"img":{"alt_text":"","src":"https:\/\/pbs.twimg.com\/media\/ELS2XD4XsAMoGzX?format=jpg&name=medium","width":350,"height":200},"classes":[]},{"id":1715,"url":"https:\/\/www.allendowney.com\/blog\/2026\/01\/31\/the-girl-born-on-tuesday\/","url_meta":{"origin":1154,"position":4},"title":"The Girl Born on Tuesday","author":"AllenDowney","date":"January 31, 2026","format":false,"excerpt":"Some people have strong opinions about this question: In a family with two children, if at least one of the children is a girl born on Tuesday, what are the chances that both children are girls? In this article, I hope to offer A solution to one interpretation of this\u2026","rel":"","context":"In \"Bayes's Theorem\"","block_context":{"text":"Bayes's Theorem","link":"https:\/\/www.allendowney.com\/blog\/tag\/bayess-theorem\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2026\/01\/c81aa262e67d9b56ecabe5664c2397cdd0375ce23e2d2c683d8d281e36c47726.png?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2026\/01\/c81aa262e67d9b56ecabe5664c2397cdd0375ce23e2d2c683d8d281e36c47726.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2026\/01\/c81aa262e67d9b56ecabe5664c2397cdd0375ce23e2d2c683d8d281e36c47726.png?resize=525%2C300&ssl=1 1.5x"},"classes":[]},{"id":1034,"url":"https:\/\/www.allendowney.com\/blog\/2023\/10\/02\/what-size-is-that-correlation\/","url_meta":{"origin":1154,"position":5},"title":"What size is that correlation?","author":"AllenDowney","date":"October 2, 2023","format":false,"excerpt":"This article is related to Chapter 6 of Probably Overthinking It, which is available for preorder now. It is also related to a new course at Brilliant.org, Explaining Variation. Suppose you find a correlation of 0.36. How would you characterize it? I posed this question to the stalwart few still\u2026","rel":"","context":"Similar post","block_context":{"text":"Similar post","link":""},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2023\/10\/Screenshot-at-2023-10-01-13-51-49.png?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2023\/10\/Screenshot-at-2023-10-01-13-51-49.png?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2023\/10\/Screenshot-at-2023-10-01-13-51-49.png?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/www.allendowney.com\/blog\/wp-content\/uploads\/2023\/10\/Screenshot-at-2023-10-01-13-51-49.png?resize=700%2C400&ssl=1 2x"},"classes":[]}],"_links":{"self":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/1154","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/comments?post=1154"}],"version-history":[{"count":3,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/1154\/revisions"}],"predecessor-version":[{"id":1162,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/posts\/1154\/revisions\/1162"}],"wp:attachment":[{"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/media?parent=1154"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/categories?post=1154"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.allendowney.com\/blog\/wp-json\/wp\/v2\/tags?post=1154"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}