Yesterday my wife, brother-in-law, and I were at the mall, watching some radio station give away a Saturn sedan. The authorities had a bucket with 70 keys in it, one of which was the key to the Saturn. There were about 70 people lined up to try their hand at picking a key and starting the car. Whoever picked the working key would win the car.
Strangely, the very first person in line--an older lady with black hair--picked the working key. The odds of this happening are quite low--about 1 in 70 I'd say, but it got my brother-in-law and I thinking about the set-up of the key drawing and the probability of winning. We wondered where in the line of 70 one should stand to maximize their odds of drawing the correct key. Since each key that didn't work would be removed from the bucket for the next contestant's pick, your odds of drawing the winning key go up as the total pool of keys decreases in number. The first person has a 1/70 chance, but that increases to 1/69 for the next person, and so on down the line to the last person who has a 100% chance of picking the right key. Unless someone else picks the winning key before them, and that's where the rub lies. Although your selection pool decreases the farther back in line you are, thereby working to increase your odds of picking the winning key, it is also the case that the odds of someone else picking it first go up the farther back in line you are.
Our questions are:
(1) Are these two factors inversely related, thereby making no difference regardless of where one stands in line, or are they not, thereby creating a position of maximal chance for success for some position in line?
(2) If the two factors previously mentioned are not inversely related, then how does one go about figuring out which position in a drawing line of the type described has the best odds of winning, for any line of length n?
I'll keep trying to sort this out, but if anyone out there either knows how to solve this or has some good ideas, I'd appreciate hearing them.
I know nothing of probability, but here's how I started out: Pretend the line has just 5 people in it. The first person has a 1/5 chance of drawing the right key. The next person in line has a 1/4 chance, provided that person 1 doesn't win. The chance of person 1 winning is 1/5, which means that there's a 4/5 chance that person 2 will have a chance to draw. So I figured that you have to multiply 4/5 times person 2's 1/4 odds to accurately find their odds of drawing the right key. This turns out to be 1/5 again, but this turns out to be a fact of these types of queues--the odds of winning are always the same for the first two in line.
However, the odds of person 3 winning are 1/3 times the odds of person 2 not winning, which is 1/3 times 4/5, or 4/15. This means that person 3 has a 26.7% chance of winning whereas persons 1 and 2 both only had a 20% chance. Position 3 is the best place to stand so far.
Person 4's odds are 1/2 for the remaining key pool odds times 11/15 for the odds of person 3 not winning. This is 11/30 odds for position 4, or 36.7% chance of winning. This is the best so far.
Person 5: 1/1 times 19/30 equals 19/30.
And now I am pretty sure my approach has gone awry. It is severely counterintuitive both that person 5 has a 63% chance of picking the key and that the aggregate percentage odds should add up to more than 100%, as they do in my approach (167%).
Last week I posted a CNN article noting that UFOs had been sighted by Mexican Airforce pilots. However, it didn't take the unbelievers long to explain away the sighting.
UFOs are, I think, an interesting subject. Probably, most of what is written is sensational and highly fictional, but it seems that people really do occassionally witness odd phenomena that doesn't appear to be explicable by recourse to ordinary channels of interpretation. I know two people, both of whom I believe to be credible, that have seen what they describe as UFOs.
The explanations I have heard for UFOs are as follows:
1. Alien aircraft
2. Experimental government aircraft
3. Rare weather phenomena
4. Demonic aircraft
I don't give any credence to 1 or 2, because I don't believe in aliens (they simply have no place in my conceptual framework), and because I don't believe the government is sufficiently competent to develop any sort of aircraft capable of the standard UFO flight abilities (i.e., rapid reversals of direction, dissappearance/reappearance, etc.) while keeping said development fully confidential.
Much more likely is 3. As in the above mentioned case, this explanation seems to make sense. My father has a book describing the many possible types of weather phenomena, and there are numerous events that I have never seen, such that should I see them, I would find them highly unusual.
4 is kind of weird. I heard a fairly disreputible but interesting speaker (Kent Hovind) say that UFOs could be means of Satanic/demonic transportation. His rationale went something like this: (a) demons exist because the Bible says so, (b) unlike God, Satan and his angels cannot be everywhere at the same time, (c) because they're not omnipresent, the demonic entities have to travel from one location to another, (d) they could use some sort of air/space craft to travel around, (e) these aircraft could be the basis of many UFO sightings.
Strange. Strange, but interesting.