Calculate this probability

[plindboe]

If a person is asked 40 questions, each of them binary(yes or no type questions), then what is the probability of getting at least 35 of them correct?

Help much appreciated.

 

[CFLarsen]

Make your question one of the 40 questions.

 

[geni]

assumming aswers are given at random then I think it is:

40C35 (0.5)³⁵(0.5)⁵

about 6*10<sup>-7</sup

 

[scribble]

If a person is asked 40 questions, each of them binary(yes or no type questions), then what is the probability of getting at least 35 of them correct?

Help much appreciated.

It depends on the questions and the person answering, of course. If you asked me 40 yes/no questions on C++ programming, I'd be ashamed to not get a perfect 40, for example.

I don't think we can answer this.

 

[TeaBag420]

If a person is asked 40 questions, each of them binary(yes or no type questions), then what is the probability of getting at least 35 of them correct?

Help much appreciated.

Unsolvable as stated. Is the person answering randomly? Does he know the answers to any of the questions? Does the questioner know what the person knows?

In the most stripped down possible version, the probability is a bit less than 12.5 %, the same as the probability of getting at least 35 questions wrong.

Remember when Steven Seagal was asked about Mt. Everest in the Glimmer Man?

 

[plindboe]

Re: Re: Calculate this probability

It depends on the questions and the person answering, of course. If you asked me 40 yes/no questions on C++ programming, I'd be ashamed to not get a perfect 40, for example.

I don't think we can answer this.

How silly of me, I should have specified. I'm talking about 50/50, like a coin toss.

I'm just having a discussion, with a proclaimed psychic, about the Larry King Live transcript below, and am interested in what the probability would be:

http://edition.cnn.com/TRANSCRIPTS/0106/05/lkl.00.html

KING: James, give us a simple example. What might be something you would ask her to do that could win her a million dollars? I know about writing it all up and everything. Give me an example. What -- you meet with her next week. You would say to her what?
RANDI: We would sit down and determine what she believes her powers are and to what extent they go, how far they will go, with what accuracy they will go. And then we would agree on a set of questions that would be asked that would have to have answers correct within a certain degree. We can do this.
KING: A question like -- like what? All right, give me an example.
RANDI: Such things as the gender of a person. That's a yes or no. That's a binary question, either right or wrong. Related, not related, old or young. And we'd have to determine what that would mean, too. A set of binary questions like this could easily decide whether or not Rosemary has the power.
KING: And how many questions would you want to have her respond to? As an example.
RANDI: It depends on what she says she can do and with what accuracy.
KING: With what you've seen tonight, what would you set up? How many questions?
RANDI: Oh, I would set up at least 40 questions.
KING: For -- and if she were right on 35, would she fail to get the million?
RANDI: You'd have to talk to a statistician about that, Larry. I'm not a statistician.
KING: Well, what would -- you give me an example.
(CROSSTALK)
What would you give the million for? What would you give the million for? Would she have to get 40 right?
ALTEA: Another wall.
RANDI: I can't talk the statistics with you. We have to refer that to our statisticians.
ALTEA: Another lock on the door. It's a very good trick, James, and I applaud you.
RANDI: There are no tricks being imposed here by me, Rosemary.
ALTEA: I think it's a really good trick. I think...
RANDI: The tricks are on your part.
ALTEA: Yeah, I think...
RANDI: I don't play tricks on people when I'm doing these tests.
ALTEA: Well...
RANDI: We agree at the test in advance, and you either pass or you don't pass. Will you try, yes or no?
KING: I'm asking a simple question, James. What is passing?
RANDI: It depends on what the test is, Larry. We have to assign the statistics to it.
KING: All right. A 40-question test, what passes? You know you're going to give away a million dollars.
RANDI: That's true.
KING: The statistician -- I assume you know statisticians. I assume you...
RANDI: No, I don't -- I know statisticians.
KING: You've never...
RANDI: I don't know statistics.
KING: So if she met with you tomorrow, you couldn't tell her what -- how many of the 40 she'd have to answer.
RANDI: Yes, I could, because I would have the right people on hand to advise me on the statistics. It's that simple.
KING: And if they told you she was 30 percent right, that would be considered extraordinarily accurate and you'd give her the million, if she had the 30 of the 40 questions right?
RANDI: If that were the statistical expectation that we could have, yes.
KING: Ah! But who sets the expectation? You can't tell us -- you have to give the statistician the expectation.
RANDI: That's right.
KING: The -- what do you ask him that he give you?
RANDI: The statistician will tell us what is significant in a set of binary questions, yes. The statistician will tell us if there were 40 questions, she would have to get x number right.
KING: I see.
RANDI: If there were 50 questions, you'd have to get...
ALTEA: I have something I don't understand.
KING: And he or -- and he or she, the statistician, would define "significant"?
RANDI: Yes, that's true.
ALTEA: I have something I don't understand. I have something here that I don't understand, because if -- James, what you're doing, and this deal is really genuine and you really expect to give it away this million dollars...
RANDI: No, I don't expect to give it away, no.
ALTEA: Well, OK -- well, you put it out there.
RANDI: I'm offering it to you.
ALTEA: OK, OK. If you -- well, then, you know, if you're seriously offering, not just to me but to anyone, I would have thought you'd really got this down. You would have had your numbers, you would have had your statisticians.
I mean, a million dollars is a lot of money. You don't just give it away and then come on TV show and say, well, it depends on this, it depends on...
RANDI: It all depends on what you claim your powers are and what your accuracy is.
ALTEA: I -- you know, I'd love you to you listen to this show. If you're recording this show, I would love you to listen to this show afterwards, because you're doing to me, to us, exactly what you're claiming I do to other people. You're doing exactly -- you are fishing, you are...

Would be great, if people could calculate the probability of 30/40 right answers as well.

 

[geni]

Re: Re: Re: Calculate this probability

Would be great, if people could calculate the probability of 30/40 right answers as well.

40C30 (0.5)³⁰(0.5)¹⁰

7.7*10^-4

It should be noted this this is the probability for exactly that number. To work out 30+ would be a different problem.

 

[JamesM]

The probability of getting at least 35 correct is 6.9e-7.
The probability of getting at least 30 correct is 1.1e-3 - that's about 0.1%.

 

[plindboe]

Re: Re: Re: Re: Calculate this probability

40C30 (0.5)³⁰(0.5)¹⁰

7.7*10^-4

It should be noted this this is the probability for exactly that number. To work out 30+ would be a different problem.

Thanks for the help, but this seems like too low a probability to me. I mean, with a coin toss, getting 30 tails out of 40 coin tosses shouldn't be that hard, just random chance

I know most people don't have a good sense of statistics, myself included, but 30/40 just seems like nothing out of the ordinary to me.

What does 40C30 mean? What does the C stand for?

 

[JamesM]

Incidentally, if you're interested in the number you'd have to get right for a certain level of significance, here they are, for 40 trials:

5%: 23/40
1%: 27/40
0.1%: 30/40

What this means is, if you were to repeat the test many, many times, and the so-called medium really was just guessing, you'd expect to get a score of above 23/40 5% of the time, and so on.

As you can see, getting more than 30 right by dumb luck would be a fairly rare occurrence.

 

[geni]

Re: Re: Re: Re: Re: Calculate this probability

Thanks for the help, but this seems like too low a probability to me. I mean, with a coin toss, getting 30 tails out of 40 coin tosses shouldn't be that hard, just random chance

I know most people don't have a good sense of statistics, myself included, but 30/40 just seems like nothing out of the ordinary to me.

The probability for gettting 50/50 is about 1/4 so don't be suppised that the probability falls off fast

What does 40C30 mean? What does the C stand for?

It means that you press the buttion marked nCr on your calculator between the 40 and 30. It is to do with a horible bit of algerbra that I haven't looked at for about 3 years. The number being calculated there is the number of posible combinations in which those answers can be arranged (note combinations not permutations).

 

[JamesM]

Re: Re: Re: Re: Re: Calculate this probability

I know most people don't have a good sense of statistics, myself included, but 30/40 just seems like nothing out of the ordinary to me.

What we have here is a binomial probability problem: you have a series of n trials, with the outcome being a simple binary yes/no, the probability of success being p.

For the binomial distribution, the expected number of successes is np, and the variance is np*(1-p).

In this case, n = 40, and p = 0.5, so:

the expected number of successes = 40*0.5 = 20
which is what you'd expect intuitively.

The variance = (40*0.5)*(1-0.5) = 10.

The square root of the variance, the standard deviation, is in the same units as the observation (in this case, correct guesses), so let's use that instead:

the standard deviation = sqrt(10), about 3.2 correct guesses.

The standard deviation measures the "spread" of the observations from its expected value, so for this experiment we might expect to get 20 right, give or take 3.2.

As you can see, getting 30 right would be more than 3 standard deviations from the expected value, which is a fairly large deviation, hence it is rather unlikely, despite what you might think.

Hope I haven't messed any of the maths up.

 

[plindboe]

Thanks alot for the answers(especially geni and JamesM).

 

[T'ai Chi]

If a person is asked 40 questions, each of them binary(yes or no type questions), then what is the probability of getting at least 35 of them correct?

Help much appreciated.

With n = 40, and p = .5, sum

nCr(n,r)*p^r*(1-p)^(n-r)

(where nCr = n!/[r!(n-r)!])

from r = 35 to r = 40

Doing that I get

.0000006913

 

[phildonnia]

Re: Re: Calculate this probability

Never mind. I type faster than I think.

 

[TeaBag420]

I especially like the part where Randi says he does and doesn't know statisticians. "Which is it?" is the question the Frog should have asked.

Also where he accepts the Frog's equation of 30/40 with 30%.

Perhaps a little too much Jesus Juice before the show.

 

[DevilsAdvocate]

I especially like the part where Randi says he does and doesn't know statisticians. "Which is it?" is the question the Frog should have asked.

Also where he accepts the Frog's equation of 30/40 with 30%.

Perhaps a little too much Jesus Juice before the show.

KING: The statistician -- I assume you know statisticians. I assume you...
RANDI: No, I don't -- I know statisticians.
KING: You've never...
RANDI: I don't know statistics.

I don;t always agree with the way Randi defends his positions, but I think in this case he was fairly clear.

Randi doesn't say he doesn't know statisticians. His answer "No, I don't" appears to be in response to King's question "I assume you..." which Randi may have been guessing would be something along the lines of "if you know statisticians, you must know something about statistics and can give me a number".

King was trying to get Randi to give a number, and Altea was accusing Randi of trickery because he wouldn't give an exact number. It appears at this point that Randi was getting a bit fed up with being pressed for a number that he couldn't provide and began anticipating King's line of questioning. To which he replied (in my interpretation "No, I don't know statistics -- I know statisticians who would help me with the statistics

It appears Randi clarifies the reply "No, I don't" when he immediately cuts off King again and says "I don't know statistics."

Also, Randi did not agree with King that 30 of 40 would be 30 percent. He didn't disagree either. Which is what would be expected if Randi doesn't know statistics.

 

[DevilsAdvocate]

Here are the probabilities of getting at least r number of right answers out of 40 questions where each question has a 50% chance of guessing right. This uses the following formula: (40!/(r!*40-r!)) * (.5^r) * (.5^(40-r)).

You may notice that it is actually MORE THAN LIKELY that you will get 20/40 questions correct. If this seems counterintuitive, consider if there are just 2 questions asked. You can get either 0, 1, or 2 right. The possible outcomes (r = right, w = wrong) are ww, wr, rw, rr. So there is a 75% chance of getting 1/2 quetsions right.

If you wanted to be over 1 in 1000 chance out of 40 questions, you would have to get at least 31 questions right. To be over 1 in 1 million would be 35 right (which by chance would be about .7 in 1 million).

The really hard part would be coming up with questions that actually have a 50% probability of being correct. For example, the ratio of males to females is not exactly 50% and can vary depending on the demographic group. So if either the psychic or the person choose the subject have any knowledge of the group from which the subject is chosen and the demographics of that group, then the test will be skewed. Therefore the questions must be agreed upon to have a 50% chance (or having a different weight of chance) and the person choosing the subject must be blind to the questions.

Here are the probabilties:

Right Chance
0 100.0000000000%
1 99.9999999999%
2 99.9999999963%
3 99.9999999253%
4 99.9999990267%
5 99.9999907149%
6 99.9999308694%
7 99.9995817708%
8 99.9978861489%
9 99.9908917085%
10 99.9660225873%
11 99.8889283113%
12 99.6786711952%
13 99.1705498313%
14 98.0761345858%
15 95.9654766124%
16 92.3070027919%
17 86.5906374473%
18 78.5204746078%
19 68.2085998686%
20 56.2685343810%
21 43.7314656190%
22 31.7914001314%
23 21.4795253922%
24 13.4093625527%
25 7.6929972081%
26 4.0345233876%
27 1.9238654142%
28 0.8294501687%
29 0.3213288048%
30 0.1110716887%
31 0.0339774127%
32 0.0091082915%
33 0.0021138511%
34 0.0004182292%
35 0.0000691306%
36 0.0000092851%
37 0.0000009733%
38 0.0000000747%
39 0.0000000037%
40 0.0000000001%

 

[TeaBag420]

I accept your corrections. I was wrong.

If Randi doesn't know statistics, how does he evaluate the qualifications of statisticians? Is there a certifying body with established standards? If he hires people to evaluate their qualifications, who evaluates THEIR qualifications?

Too much thought.... must get Jesus Juice.

 

[DevilsAdvocate]

I accept your corrections. I was wrong.

If Randi doesn't know statistics, how does he evaluate the qualifications of statisticians? Is there a certifying body with established standards? If he hires people to evaluate their qualifications, who evaluates THEIR qualifications?

Too much thought.... must get Jesus Juice.

That opens a differnt philosophical can of worms: how do you know what is true? Why do you believe what you believe? When do you accept a belief as true? Tough questions.

We have seen a number of different answers to the same question: some wrong, some right, some right but actually answer a different question. Statistics can be proven mathematically, but who knows if the math is correct?

You might say that Randi has (gulp) FAITH that his satisticians are correct. Actually, the staisticians' results can be verified by proven repeatable results. I'm sure that you can find statisticians that you can trust to be accurate -- it's not really that hard (and I'm not saying that you should trust my numbers -- I could be wrong).

Anyway, for the JREF challenge it doesn't matter much whether the statisticians are right or wrong as long as the protocol is agreed upon. If Randi's stats people say getting 21 out of 40 is acceptable and the psychic gets just a little lucky, then so be it. JREF loses 1 million$ but most people probably won't acept this as paranormal. The real problem would be if JREF's stats people set the bar impossibly high, which would be cause for believers to acuse JREF of being unfair.

In my opinion, I think the bar for "paranormal" should be set very high. There should be a clear distinction between "incredible chance", "amazing capability to deduce information (like the guy Sherlock Holmes as based on or certain cold readers) and truly "paranormal".

I can understand that a psychic's ability could exist that would be somewhat fuzzy, that would sometimes work and sometimes not, that would require the right concentration and conditions. I can spell and I can type, but my spelling and typing in this post probably aren't perfect. I can also understand that a psychic's ability may be determined on what they can "see".

For example, if I calimed to be abl to "see" certain things about a person (which happens to be because I have a fuzzy, grainy photo of them) I could tell you certain things about the person. But if I had a presceipted list of questions, the answers to those questions may not be available in the photo that I see. So I might fail the test miserably, but actually have the abilty to "see" certain things about the person. So is the test proposed by Randi really fair?

This is one reason why Randi and JREF have to make different protocols for each cliamaint, and why they are so difficult. In my opinion, to prove paranormal abilty, a psychic should be able to tell you a certain number of things about a person just as I would if I looked at a fuzzy, grainy photo of the person. If I told you some things about a person in a fuzzy, grainy photo then I might get some things wrong, but I should be WAY beyond probability. After all, I really can "see" them (in the bad photo). And I should be able to prove my ability to describe things about a person in a fuzzy, grainy photo WAY beyond probabilty with repeatable results. I think I can do this with lokking at poor photos. Can psychics do this by paranormal means? Not that I've seen.