Quick Probability Questions for Interviews

1) If cor(a,b)=0.5 and cor(a,c)=0.0, what is the range for cor(b,c)

2) A deck of pokers. Three choices: A: 26 black, 26 Red; B: 13 black, 13 Red; C: random 26 card from the deck. Take the first two cards, if same color, the win $1, otherwise lose$1. Which deck is best for you if you are playing? Why?

3) What is the probability of a random generator generating 10 consecutive numbers in ascending order (assume it is a perfect random generator)

Update (24/12/2012):
Solution posted by AH in comments! Solution explained in detail by me. Thanks

1. 1) It will be like solving matrix
[\sigma_a^2 0.5\sigma_a\sigma_b 0 \\
0.5\sigma_a\sigma_b \sigma_b^2 cor(b,c)\sigma_a\sigma_b \\
0 cor(b,c)\sigma_a\sigma_b \sigma_c^2]
such that each row and column sum is 1

Was not able to solve it completely

2) 26B 26R Probability = 25/51
13R13B 12/25
26 random number
Let x R and 26-x B cards are choosen
Now x can be uniformly between 0 to 26
Probability = \integral_{0}{26} (x(x-1)+(26-x)(25-x))/(26*25*26) dx
Probability = 0.653
Deck 3 is the best

2. 1.) Correlation matrix should be non negative definite => determinants at all levels should be >= 0 => x^2 < 3/4

2.) Expected Winnings:
Choice A: ((1)*(25/51)) + ((-1)*(26/51)) = -1/51
Choice B: ((1)*(12/25)) + ((-1)*(13/25)) = -1/13
Choice C: Same as A (Sampling 26 random cards and then taking the first two is same as just taking the first two from the whole deck)

3.) 1/10!

3. @AH.. Thanks for the correct solutions to all the three problems.

Q.1

A more detailed solution of first question:
The correlation matrix for a, b, c is as follows:
first row as (1.0, 0.5, 0.0)
second row as (0.5, 1.0, x)
third row as (0.0, x, 1.0)

has to be positive definite
Hence, 1*(1-x^2) - 0.5*(0.5) >= 0
Hence, x^2 < 0.75

Q.2

A more mathematical explanation for the Choice C of the second question is "Total Probability Theorem".

1. Q1) why is variance of a, b, c =1

2. without loss of generality you can take mean=0 and variance=1. This is because correlation is invariant over location and scale transformations. Correlation between (x,y) =
Correlation between ((x-a)/b , (y-c)/d)

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Fraction Brainteaser

Source:
Sent to me by Gaurav Sinha

Problem:
Siddhant writes a Maths test and correctly answers 5 out of 6 Arithmetic questions and 20 out of 28 Geometry questions. In total, Siddhant scores 25 out of 34.

Vaibhav writes another Maths test and correctly answers 20 out of 25 Arithmetic questions and 6 out of 9 Geometry questions. in total, Vaibhav scores 26 out of 34.

Note that
a) Vaibhav scores more than Siddhant
b) Siddhant score better than Vaibhav in both individual topics - 5/6 > 20/25 and 20/28 > 6/9

How is it possible?