Probability
Suppose you roll 2 different die, each with 3 sides (with values 1, 2, 3). What is the probability that the sum of the numbers rolled is <= 4?
2/3 = 66.6%
How do you calculate the binomial coefficient nCk (n choose k)?
n!/(n-k)!k!
An atom has a 10% chance of decaying in one day. What is the probability of an atom decaying over two days?
1 - (p(surviving))^2
1 - (0.9)^2
0.19 = 19%
Find the eigenvectors and eigenvalues of
[-6 3 ; 4 5]
Eigenvalues: 6, -7
Eigenvectors: [1 ; 4] [-3 ; 1]
What is the mathematical definition of two events being independent?
P(A&B) = P(A)*P(B)
What is the probability that exactly k molecules out of n are in a particular box at a particular timestep?
nCk * deltak * (1 - delta)k
An atom has a 15% chance of decaying in one day. What is the probability of an atom decaying over seven days?
1 - (p(surviving))^2
1-(0.85)^2
0.2775
27.75%
Find the eigenvalues and eigenvectors of
[ 5 -3 ; 2 1]
Eigenvalues: 3 +/- root(2)*i
Eigenvectors: [ 3 +/- root(2)*i ; 2]
There are six cards face down. Two of them are red, while the remaining four are black. I pick three of them at random. What is the probability that I end up picking at least one of the red cards?
80%
What is the probability of never rolling a 1 on a 10 sided die?
36.7%
An atom has a 10% chance of decaying in one week. What is the probability of an atom decaying over one day?
(1-p)^7 = 90%
p = 1.49%
Find the eigenvectors and eigenvalues of
[1 2 1 ; 6 -1 0 ; -1 -2 -1]
Eigenvalues: 0, 3, -4
Eigenvectors: [-1 ; -6 ; 13], [-2 ; -3 ; 2], [-1 ; 2 ; 1]
I shuffle a deck of cards and draw a hand of 7. What are the chances that I get at least one spade?
88.5%
Between 4 items, how many pairs of 2 can be created?
12
An atom has a 30% chance of decaying in one day. What is the probability of an atom decaying over one hour?
1.48% per hour
I shuffle a deck of cards and draw a hand of 7. What are the chances that I get at least one spade?
26.37%
Out of 8 elements, how many groups of 3 can be created?
56
An atom has a 30% chance of decaying in one hour. What is the probability of an atom decaying over one minute?
0.59% per minute