A friend of mine was bragging about how good he was at shooting. He had recently joined a shooting club, where the club average for missing the target was 1%. My friend was bragging that despite taking 70 shots at the target, he hadn't missed yet. I tried telling him that wasn't so impressive, but he didn't want to hear it.

Which of us was right? Is he really that great?

Not that impressive. If the average shooter misses only 1%, and if "your friend" is an average shooter, then he probably would not have missed. (1% of 70 is only .7, or less than one shot.)

ReplyDeleteI disagree. If he were an average shooter, he would have a 50.5% chance of missing the target by his 70th shot. Not something to brag about, maybe, but the statistics say he is better than the average gunman at the club. Here's how:

ReplyDeleteThe chance of an average shooter missing the target is 0.01, and so the chance of an average shooter hitting the target is 0.99. Therefore, the probability that he would have missed the target at least once after 70 shots is

P(not perfect) = 1 - P(perfect)

The probability that he could have shot 70 times without missing is

P(perfect) = (0.99)^70 = 0.495

p(not perfect) = 1-0.495 = 0.505

You can't handle probability problems like that by just adding together the probabilities for each time an "event" occurs. That would be like saying you have a probability of 100% of heads after two flips of the coin, which we all know is not true.

I solved the problem on my own and came up with the same answer as Abe. The likelihood of going 70 shots without a miss if the likelihood if each individual shot has a 99% success probability is about 50%, meaning that if all shooters have that same capability, about half will get to 70 without a miss, and about half won't.

ReplyDeleteI disagree with everybody's method, but not with the answer. I say that it is too early to determine whether or not he is average or above average

ReplyDeletebecause the average is 99% accuracy (99/100) and he has only taken 70 shots, he really actually needs 30 more shots before anything is conclusive. if he misses one of the next 30, he is average.

another way of looking at this is through the fact that if you take one shot and hit, you have 100% accuracy. but you have only taken one shot. to me, taking one shot is just as inconclusive as only taking 70.

Next, by using the probability theorem, you imply that his shot is simply a statistic. which doesnt take into account the human element. plus, even if you have a 1% chance of an event taking place (hitting the target), thats not to say that it could still happen.

In conclusion, my decision stands that the data so far is inconclusive, therefore making him an average shooter until he proves that he is extrordinary.

nick your explanation is totally without merit. for any number of events, you can define a confidence level. true, the more shots he takes, the higher the confidence, but where the hell do you get 30 more shots as some magic number>

ReplyDeletebut andy, I think you're forgetting about the human element. don't ever forget the human element.

ReplyDeleteI think all of you have the right idea, but Nick is right when he says it's too early to tell if our shooter is average or not. Yes, the probability is about 1/2 that even if he is average, but we don't know if he is average or not.

ReplyDeleteOf course, the next question is at what point could we say he isn't an average shooter? ;-)