In the movie "What's Your Number?" a young woman Ally Darling (played by Anna Faris) obsesses about the number of men she has slept with in her lifetime. She reads in Marie Claire magazine that "the average number of lovers women have in their lifetime is 10.5". Further, according to a "Harvard study", 96% of women who have been with 20 or more lovers can't find a husband. One of Ally's friends summarizes, "When you are too sexually available it messes with your self-esteem. Next thing you know, you are 45 with no self-respect and no husband".

Ally's current Number is 19, almost twice the average. She is 1 away from the dreaded 20, and with it certain spinsterhood. But if she can find a husband among her many ex-boyfriends, her Number will remain 19 and all will be well. This silly premise provides the plot structure for the movie.

Topic #1 - Fun With Averages

Ally and her friends compare Numbers. Katie's is 4. Daisy's is 8. Jamie's is 6, but this is increased to 9 after everyone agrees that inserting "more than the tip" counts as sex. Sheila's is 13, whereupon the girls call her "Whore!" and "Slut!". Ally's is 19, causing Sheila to welcome her fellow slut with, "Hello, friend".

What is the average Number among these women? Calculating the average (i.e. arithmetic mean) produces (4 + 8 + 9 + 13 + 19) / 5 = 10.0. Ally's friends are close to the supposed national average of 10.5.

The average Number for men must be higher than the average Number for women, because men are hornier, right? Wrong! Even if men are hornier, their average Number must be exactly the same! We'll see why in a moment.

A (real) study said that "Males 30-44 report an average of 6-8 female sexual partners in their lifetime" and "Females 30-44 report an average of 4 male sexual partners in their lifetime". Someone is lying! Either the men are over-reporting, or the women are under-reporting, likely both. To see why the average Number must be the same for women and men, consider the following example.

There is an isolated village containing 10 women, and 10 men. To keep things simple, women sleep only with men, men sleep only with women, everyone agrees what is considered sex, everyone has a perfect memory, and everyone reports their Number honestly. 9 of the women, all named Mary, have slept only with their husbands. The 10th woman, Britney, is a slut and has slept with all 10 men. 9 of the men, all named Roger, have slept with their wife, plus Britney. The 10th man, Gus, is unmarried and has only slept with Britney.

What is the average Number for the women? It is (1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 10) / 10 = 1.9. The middle number (the Median) is 1, but busy Britney single-handedly almost doubles the average to 1.9.

What is the average Number for the men? It is (2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 1) / 10 = 1.9. The median is 2, and gloomy Gus brings the average down a bit to 1.9.

The average Number for the woman and the men are exactly the same. They must be! Consider this diagram, where a line connects each woman on the left to her sexual partner(s) on the right, and vice-versa. Now we see why the average Numbers must match. It takes two to tango -- each line has two ends. Each liaison adds 1 partner-count to the left side and 1 partner-count to the right side. So the Totals must be the same, and, because there are equal numbers of women and men, the Averages must be the same also!

The Averages are the same, but as the example illustrates, the Average can disguise wildly different distributions. The women of the village have a very uneven distribution, with their Average very different from their Median. The men of the village have a more even distribution, with their Average very close to their Median. So neither the Average nor the Median is a perfect measure. The Average doesn't tell you anything about the underlying distribution. The Median tells you nothing about what's going on at the low and high ends of the distribution. For cases like these, I prefer the lesser-known Geometric Mean, which is the product (rather than sum) of all the numbers, raised to the 1/10th power (rather than divided by 10). The geometric mean Number of the women is 1.3, and the geometric mean Number of the men is 1.9. These seem closer to the intuitive "average" obtained by just looking at the numbers.

Topic #2 - Picking The Best Husband

When the movie opens, Ally is already desperate and is willing to settle for any adequate husband. But let us play a different game.

Suppose that we could rewind time for Ally. We tell Ally that we are going to present 20 guys to her, one at a time, and she has to pick which one she wants as a husband. She has no idea what kind of guys we will present. For each guy, she can choose him for her husband, or pass. Once she passes on a guy, he is lost to her forever (unlike in the movie, she can't go back to him later). If she passes on 19 guys, then she must pick the 20th guy to be her husband. Being young and naïve, Ally wants to choose the very best guy, the pick of the litter, not just a good guy. How should Ally do this? What are her chances of successfully picking the very best guy from the 20 available?

If she covers her eyes, and simply chooses one of the guys at random, Ally chances of picking the best guy are only 5% (1/20). The math behind it is complicated, but there is a much better method, and it increases her chances of picking the best guy to a decent 37%. The technique is simple. Divide the number of guys by E (the natural logarithm) and round to the nearest whole number: 20 / 2.718 = 7. So Ally should carefully examine the first 7 guys, but pick none of them. After the first 7 have gone by, Ally should pick the very next guy who is better than all of the first 7. Using this technique, Ally maximizes her chances of picking the best guy, and her chances are (1/E) or 37%.

The math is valid for any collection of 10+ objects, where the range of objects is unlimited and the objects have not been seen before. Try it with a random gallery of pictures. Look at 10 pictures one at a time, and try to pick the hottest guy (or gal). The math says you should examine the first 3 pictures, and then pick the very next picture that is better than all of the first 3. It works pretty well!

Ms. PartyPilgrims