First, I'm afraid I fell victim to one of the classic blunders—the most famous of which is never get in involved in a land war in Asia, but only slightly less well known is this: probability is not additive (as a couple astute Times readers have now pointed out--before going further, you might want to read my Frankie the Fixer puzzle in the left sidebar of THIS PAGE).
For example, if you have a 4-sided die, there's a 1/4 chance of rolling a specific number first try. If you roll again, is there a 2/4 chance, and after five rolls a 5/4 chance? Nope. Instead the math goes like this:
• one roll of a d4: (4^1-3^1)/4^1=1/4=0.25
• two rolls of a d4: (4^2-3^2)/4^2=7/16=0.44
• three rolls of a d4: (4^3-3^3)/4^3=37/64=0.58
• four rolls of a d4: (4^4-3^4)/4^4=175/256=0.68
• twenty rolls of a d4: (4^20-3^20)/4^20=1096024843375/1099511627776=0.97
APPROACHING BUT NEVER EQUALING 1/1 OR 1.0 OR 100%. (There's never a sure bet...)
And so my approach, stated word-for-word in the puzzle Frankie the Fixer, of "adding 1% to each roll's chance of being boxcars" is extremely misdirecting—in fact, I misdirected myself—it isn't as simple as summing 1% across the number of rolls and adding the result. That's exactly the same as adding 0.25 every time you roll a d4, which leads to the erroneous probability of 5/4 after five rolls.
Instead—as a couple readers pointed out—the trick is making the 1% additions ALONG WITH EACH 2xd6 ROLL'S 1/36 CHANCE, thus looking at each roll as a 0.028+0.01=0.038 probability of success. The gist of this rambling (but I hope definitive) explanation is that—yes—it takes 18 rather than my stated 16 throws for Frankie the Fixer to break the
magic 0.50 probability.
Here are answers to yesterday's fiendish puzzles:
1. The trick is to convert everything into runners on base per out. In this case, Rivers' cutter earns .168 on-base/out, and his fastball earns .252 on-base/out. The slugger is trickier. A .345 ob% for cutters means that each at-bat he has a .655 chance of being out, and so .527 on-base/out. Facing fastballs he earns .453 on-base/out. The total on-base/out of a cutter is the average of pitcher and hitter—0.348 on-base/out. The total on-base/out of a fastball is 0.353 on-base/out. So Rivers' best pitch remains his best pitch, the cutter.
2. This is a twist on Martin Gardner's famous gender problem. First, combining birth order with gender means with two kids you could have BB, BG, GG, or GB. Now, Imagine the number of distinct possibilities with the calendar:
• If you FIRST have a boy-on-a-day-containing-a-1, you could have a boy or a girl second, on any of the 31 days, for a total of 62 possibilities, 31 of which are two boys. Cool.
• And the same is true if you SECOND have a boy-on-a-1: 62 possibilities of which half are boys. Only, 13 of these "new" possibilities aren't distinct. You already included boy-boy on every day containing a one. So instead of adding 62 more, distinct possibilities, this adds only 49 new possibilities, of which only 18 are two boys.
• So add up all the possibilities for two boys: 31+18=49. And add up all possibilities:
62+49=111. There's a 49/111=0.44 probability that both kids will be boys.