What are the Chances?

2 May

A while back Judy on Zebra Sounds linked to a “Brain Test” What Sex is your Brain. While interesting, it is certainly not nearly long enough to be statistically significant, and like most pop psychology, it simplifies things to the point of absurdity. For example, being empathetic and recognizing the signs that would trigger empathy are two distinct items, as, I think, anyone with Asperger’s Syndrome or who have dealt with them would know.

There was one section, where my score I found particularly odd – that is identifying emotions with eyes. I got 5 of 10 correct. There were only 2 or 3 that I looked at and my response was represented by one of the 4 choices. This lead me to the question “What is the probability of getting 5 right by chance?” followed by getting 5 right given 2 or 3 known. I am only going to address the first question.

If this was a binomial problem with a 50:50 outcome then is because a simple combinational problem. For example what are the chance of getting 5 heads in ten tosses of a fair coin?

Letting k be the number of heads and n be the number of tosses the formula is:

n!/(k! * (n-k!))

since we don’t care about the order. The ! is factorial, meaning 3! equals 3 x 2 x 1; and by definition (IIRC) 0! = 1

The following table shows the different number of combinations:

Combination
Number wrong
Combinations
0
1
1
10
2
45
3
120
4
210
5
252
6
210
7
120
8
45
9
10
10
1

The sum of all the combinations is 1024, so the chance of tossing 5 heads in 10 flips is 252/1024 = 0.24609 or approximately 25%.

For this test, it is more complicated since there is a 75% chance by guessing of getting the wrong answer; that is getting all 10 wrong would be 0.75^ 10. Getting 5 right and 5 wrong would be 0.75^5 x 0.25^5 x 252. There is only one combination where they are all right or wrong but 252 where half are right and half are wrong.

Probability
Number wrong
Combinations
Individual Prob
Combination Prob
0
1
9.537E-07
9.537E-07
2
45
2.861E-06
2.861E-05
3
120
2.575E-05
3.090E-03
4
210
7.725E-05
1.622E-02
5
252
2.317E-04
5.840E-02
6
210
6.952E-04
1.460E-01
7
120
2.086E-03
2.503E-01
8
45
6.257E-03
2.816E-01
8
45
1.877E-02
2.816E-01
9
10
1.877E-02
1.877E-01
10
1
5.631E-02
5.631E-02

Interestingly, the chance of getting 5 right by randomly picking, apx 6%, is very close to getting them all wrong.

I will delve into the probability of getting 5 answers correct given 2 or 3 answers are know later.

Note: I see WordPress removes the center align tags in the tables – Nice.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: