What are the odds of a tie and the distribuion in this situation?

Factual Questions
1

Anyone with a good knowledge of statistics will probably find this to be a ridiculously simple math problem. My knowledge of statistics is beyond rusty, alas, so I turn to Dopers to solve this one:

As part of some birthday party activities, I’m going to administer a 15-question multiple choice quiz to about 40 people. All questions will have two possible answers. People who get the most questions correct will get small prizes.

Assuming that everyone answers the questions on a random basis, what will the distribution of responses look like? The expected value is 7.5 correct answers, but there should be a few people who get 10 or 12, maybe even more, correct, no?

I just want to be prepared with enough prizes, so what is the likelihood (again, assuming random chance) of 15 correct answers? 14? etc.

2

I’m not sure that calculating it on the basis that people’s answers will be pure guesses is going to be that helpful, but it depends on the questions, I suppose. Anyway, under that assumption I get:

Number of distinct, equally likely ways of answering the quiz: 2^15
Number of ways of answering n questions correctly: (15 choose n) = 15! / n!(15-n)!

Hence p(n correct answers) = 15! / ( 2^15(n!)(15-n)!)


no correct	% of players	no of players
0		0.00%		0.0
1		0.05%		0.0
2		0.32%		0.1
3		1.39%		0.6
4		4.17%		1.7
5		9.16%		3.7
6		15.27%		6.1
7		19.64%		7.9
8		19.64%		7.9
9		15.27%		6.1
10		9.16%		3.7
11		4.17%		1.7
12		1.39%		0.6
13		0.32%		0.1
14		0.05%		0.0
15		0.00%		0.0
3

Is the random-answer assumption reasonable? Don’t the questions require knowledge? Anyway , here’s what I get with simulation:

Winning Score
0.1% 15
1.9% 14
11.6% 13
37.2% 12
40.6% 11
8.5% 10
0.1% 9

56.9% chance of single winner
22.9% chance of two winners
10.7% chance of three winners
5.1% chance of four winners
2.4% chance of five winners
1.1% chance of six winners
0.5% chance of seven winners
0.4% chance of eight winners or more

There will usually be a single winner, but you need to allow for as many as eight tied for the lead. Best may be to enforce a minimum score.

If only 11 or better qualifies
56.3% chance of single winner
21.6% chance of two winners
8.9% chance of three winners
3.3% chance of four winners
1.0% chance of five winners
0.3% chance of six winners or more
8.6% chance of random draw among 10-scorers

If only 12 or better or undisputed champ qualifies
56.9% chance of single winner
8.3% chance of two winners
1.4% chance of three winners
0.2% chance of four winners
0.02% chance of five winners or more
33.2% chance for random draw among high-scorers

4

You should also consider that there may be some who will reject both options “none of the above” or even accept both options.

5

One way to get around this is to have a tie-breaker question if necessary. To make sure that multiple people can’t get it right, the answer to a question can be a number (“how many miles is it from LA to Chicago?”); people can write down their best guess, and the closest guess (or the three closest guesses, if you have multiple prizes) are the winner(s).

6

Thanks guys.

I knew there would probably be questions about “is it reasonable to assume random chance” but didn’t want to get into it. Obviously, if there are one or more persons with enough knowledge to do better than random chance on several of the questions, that will produce non-random outcomes. But given the questions, I think that’s unlikely. Also, it would be a much tougher problem to state - I’d have to make a variety of assumptions that I have no evidence for, a la “assume 20% of the people have a 2/3 chance of answering 7 of the questions correctly”.

Anyway, if I prepare for 5 winners, I should be safe. And while I had thought about having a tie-breaker question, I hadn’t thought of doing something as effective as MikeS’s suggestion - thanks, that’s perfect!

7

I’d have said it would be hard to come up with 15 questions that a group of 40 people would have no knowledge about - at least without the questions looking strange and provoking a rather strong “what the heck is this?” reaction from the group.

8

Some guests may eat the test … it’s been known to happen …

9

Well, they will be told beforehand to put their mobile devices down so they can’t Google anything.

The event is a belated 60th birthday party for my husband - here is a sample question:

Among his lesser known endeavors, actor, producer and director Tom Hanks provided the voice of Woody in the 2011 Pixar short Hawaiian Vacation. Is Tom Hanks older than Cairospouse, or younger?

The correct answer is Tom Hanks is older; their birthdays are just a few weeks apart.

Since most people, at least not my friends, don’t go around with Tom Hanks’s exact date of birth in their minds, I figure it’s gonna be kind of a random chance sort of thing.

ETA: The Hawaiian bit probably looks weird, but it is a combo genuine Hawaii party and fake “Tiki culture” party as well as a birthday party. My co-hosts were both born and raised in Hawaii.

10

Don’t assume that the answers will be random just because people don’t know the answers. Often, people will try to avoid patterns or come up with other schemes when they don’t know how to answer. For example, they will try to avoid choosing the same answer more than two or three times in a row, even though you would expect some clustering in a truly random sample.