A Pop Quiz About Percentages and Probabilities
Some of these are purely math problems. Others have confounding issues beyond pure math. In other words, they are trick questions. Good luck.
1. If you were at a party with 30 people, assuming no one was born on February 29th, what are the chances two of them share the same birthday?
There's an approximately 71% chance two people out of 30 will have the same birthday. This may seem unlikely, but it's true. Mathematically it's easier to figure out the possibilities of non-matching dates than matching dates. So the probability of a match is the inverse of the non-matching probability.
In other words, what is the probability that it is NOT true that at least two of them have the same birthday? Or: what is the probability that all of them have different birthdays? When we have computed that, for example we might find the answer is 0.4 (40%), then the probability we look for is 'the rest', which in this case would be 1 - 0.4 = 0.6 (60%).
Let's look at the probability for three people, Tom, Dick and Harry. The first person, Tom, equals 1 because he always matches his own birthday so the probability is 100%.
Tom ---> always fits = 1
Thus the probability for three different birthdays NOT matching is 1 x 364/365 x 363/365 = 0.99. Which means the probability that at least two of them have the same birthday is approximately 1 - 0.99 = 0.01 (1%).
Now we know how to do the problem with 30 people NOT matching:
1 x 364/365 x 363/365 x ...etc... x 336/365 = 0.29. Which means the probability that at least two of them have the same birthday is approximately 1 - 0.29 = 0.71 (71%).
This is a pretty unwieldy formula, so mathematicians have a shorter way to write it comparing any number of people (n). First off, we don't need the "1x" at the beginning so we're left with the rest. If you know permutation notation, you can write this formula as:
(365_P_n)/(365^n) That's the same as 365! / ((365-n)! * 365^n)
Rather than explaining what all that actually means, I'll leave it to the math experts.
2. Assume a hypothetical disease, call it Terry's Syndrome (TS). On average, one in 1,000 people get TS. The standard test for TS has a 0.9% false positive rate. The test also has a 0% false negative rate, if the test is negative you do not have it. If you test positive, what is the probability you actually have TS?
There's a ten percent (10%) chance you have TS when testing positive. Since one in 1,000 people have TS and will have a true positive result (remember, there are no false negatives), that's a 0.1% true positive rate. Compared to the 0.9% false positive rate, false positives outnumber the true positives 9 to 1. So one in 10 total positive results, or 10%, are true. All the same, 99% of the test results will be true negatives and only 1% will be true and false positives.
3. Say you were a baseball manager in the American League, so you had a designated hitter. Say you had nine really good hitters and you wanted to try every possible lineup combination to see what work best. The baseball season is 162 games. If you began on opening day of the 2008 season, when would you run out of possibilities?
You could have a different lineup every game until the end of the season, in the year 4247. That's because it would take 2,240 years to exhaust every possibility. Don't buy it? Consider: there are 9 different possible leadoff hitters multiplied by 8 possible players left to hit second, multiplied by 7 remaining number three hitters, times 6 possible cleanup hitters... etc. The math would be:
9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880 possible lineups
Divide that by the 162 games in a season and you wind up with 2,240 years. Not counting playoffs or spring training, that is.
It would not be in the year 4248 (as in 2008 + 2240) because you must include the year 2008 in the addition as you began at the start of the year. In effect you begin after the end of the 2007 season and run out 2,240 seasons later, at the end of the 4247 season.
4. 850 people accidentally get a 15% lethal dose of some toxic substance. How many people would you expect to die from this?
Zero, nobody will die. As they say, the poison is in the dose. A 15% lethal dose isn't lethal no matter how many people were exposed. The dose is for each person, there is no cumulative effect. People consume toxins daily, lots of them. But the doses are too small to have any effect.
5. You're at the craps table in Vegas and have rolled 5 straight sevens. What are the odds of rolling a 6th straight seven?
Five to one against. This one's kind-of a gimme. On any one roll of the dice there's 36 combinations of numbers from two to twelve, of which 6 total seven. So the odds are 30 to 6, or 5 to one. What happened the previous five rolls make no difference as each roll is entirely independent. On the other hand, the odds of rolling six straight sevens is 15,625 to one. But that's before you roll the first one.
6. What percentage of the fruits and vegetables the average American consumes are organic?
One hundred percent (100%) of produce eaten is organic. There's actually no such thing as non-organic produce. It's the farming method that's "organic," not the produce. All fruits and vegetables grow from the same seeds with the same genetics which grow in the same way. They are practically identical in every way. How the nutrients are added to the soil makes little or no difference.
Think of it this way, is there really any difference between a hand-made chair and a machine-made chair of the same material and design? Other than the price, that is.
7. You are given a choice of three secret prizes in boxes, knowing two are worthless and one is a winner. After making a choice one of the unchosen boxes is opened revealing a worthless prize. You are then given the option of switching your choice between the two unopened boxes. Should you do it?
Yes, you should switch. It increases your chances of winning from 33% to 67%. All is explained here.
8. If 87% of the leading scientists and 43% of the general population accept the new Colon Hypothesis of Gravitational Vorticity (CHGV), what are the chances that it's correct?
There's no way to tell because scientific truth isn't established by voting, but by factual evidence. CHGV might be true if nobody believed it or it may be wrong even if everyone believed it.
9. If you neutered 50% of male cats, what percentage reduction in cat reproductive capacity would you get?
Maybe a five percent (5%) reduction in cat reproductive capacity. This is not so much a math question as a biological issue of cat mating proclivities. Because cats are not monogamous, one male cat can impregnate many females. (Btw, one cat litter can actually have multiple fathers.) Theoretically, you could eliminate half the males and the remaining males could still impregnate all the females, so there'd be no decline in reproduction at all.
In practice, you get about a 5% reduction from neutering 50% of the males. You don't get any significant reduction until you reach around a 75% rate. On the other hand, there is a direct one for one reduction from spaying females: 50% fewer females means 50% fewer possible litters, no matter how many males are out there. (Unless there are none, in which case all bets are off.)
10. If you can fool 60% of the people 100% of the time, and 100% of the people 70% of the time, how likely are you to be elected to Congress?
You have a 0.00047% chance of being elected to Congress. That is if you forget about all the business of fooling people and just divide the number of Congressional seats by the number of eligible Americans. That's your theoretical chances.
© Terry Colon, 2007