I’m kicking off a new feature on this blog with a reader quiz about probability. Read on if you’d like to test your skills against fellow Numbers Guy readers.
I’m planning to regularly interview authors of new, interesting books about math and statistics. First up in this series is Leonard Mlodinow, a former writer for “Star Trek” whose inclination to muse about statistics and randomness led him to take up his son’s dare to try to shoot an apple off his head with bow and foam-tipped arrow. Some might call that eccentric; to Mr. Mlodinow it’s a Bernoulli process, and he wonders how random error might affect his shot. (His son,
Nikolai Nicolai*, was unscathed.)
Mr. Mlodinow, a visiting lecturer at Cal Tech and co-author of “A
Brief Briefer* History of Time” with Stephen Hawking, peppers “The Drunkard’s Walk,” set to be published next month, with dozens of examples, ranging from historical to personal to newsy. They serve to sketch an engaging history of probability and statistics, and to bolster his underlying thesis that the randomness that afflicts a sot’s amblings is pervasive in our lives. Furthermore, Mr. Mlodinow argues, we often act as if under the influence, failing to recognize the randomness in patterns and the patterns in randomness.
I’ll be interviewing Mr. Mlodinow by email, and publishing our exchange on the blog. If you’d like to suggest questions, please email me. For a warm-up to the interview, and to give a taste of what the book covers, the following is an eight-question quiz derived from examples in the book.
All entries may be quoted by name, unless you specify otherwise. Please email me (firstname.lastname@example.org) your responses by Thursday, April 10. No explanation is needed, though feel free to provide commentary on your answers if you’d like. (I’m closing comments on this post to ensure no one posts an answer key.) The deadline has passed. Thanks to everyone who entered. I’ll report back on the results and explain the answers next week. You can read the results and explanations of the answers, along with a lively discussion about the quiz.
1. Suppose 1,000 athletes are tested for drugs. One in 10 have used the drugs, and the test has a 1% false-positive rate (and the false-negative rate is negligible). If an athlete from this group tests positive, what is the probability that she has used the drugs, to the nearest percentage point?
2. You know that a certain family has two children, and that at least one is a girl. But you can’t recall whether both are girls. What is the probability that the family has two girls — to the nearest percentage point?
3. You know that a certain family has two children, and you remember that at least one is a girl with a very unusual name (that, say, one in a million females share), but you can’t recall whether both children are girls. What is the probability that the family has two girls — to the nearest percentage point?
4. In baseball, suppose the American League champion is better than the National League champion, such that it has a 55% probability of winning each game against the NL champ. Then the NL champ nonetheless will win a best-of-seven-games series four in 10 times. What is the smallest odd number, X, for which a World Series between these two league champs that is best-of-X will ensure that there’s a 95% probability of a just result — the superior AL champ winning?
5. Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind each of the other two, a cow. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, to reveal a cow. He then says to you, “Do you want to change your choice to door No. 2?” Is it to your advantage to switch your choice, assuming you prefer cars to cows?
6. Suppose that 60% of eligible voters in Scranton, Pa., prefer Hillary Clinton to Barack Obama. If you randomly select five of these prospective voters and they tell you their true preferences, what is the probability that exactly three of these five support Clinton?
7. In a large set of financial data — for instance, the revenue of Fortune 500 companies in each of the last 50 years — what proportion of the financial figures should you expect to begin with the digit 1, to the nearest 5 percentage points?
8. Which number is greater, the number of six-letter English words having “n” as their fifth letter or the number of six-letter English words ending in “ing”?
Further reading: I’ve written previously that any number can seem to hold special properties because of randomness, and questioned the randomness of the iPod Shuffle. For more reading on numbers, see my latest reading list. In this blog’s infancy last year, I posted a list of recommended books, and also conducted a game-theory contest.
*Corrections: Leonard Mlodinow is co-author of a “A Briefer History of Time,” and his son’s name is Nicolai. This blog post initially misstated the name of that book and misspelled his son’s name.