Probability Flashcards
p(E) = ?
|E|/|S|
p(E | F)
p(E n F) / p(F)
P(!E) = ?
1 - p(E)
Independence of E and F
p(E n F) = p(E) * p(F)
Independence of p(E | F) = ?
p(E)
Bays Thereon p(F | E) = ?
( p(E | F) * p(F) ) / ( ( p(E | F) * p(F) + p(E | !F) * p(!F) )
Probability of k successes out of n trials
p = probability of success
q = probability of failure
C(n,k)(p^k)(q^n-k)
p(E1 u E2)
p(E1) + p(E2) - p(E1 n E2)
probability 5 card poker hand has ace of hearts
1/52 + 1/52 + 1/52 + 1/52 + 1/52 = 5/52
probability 5 card poker hand does not have queen of hearts
1 - (1/52 + 1/52 + 1/52 + 1/52 + 1/52) = 47/52
probability 5 card poker hand contains 2D, 3S, 6H 10C, KH
1/C(52,5)
probability that five card poker hand contains at least 1 ace
1-(C(48,5)/C(52,5))
probability that a five card hand contains exactly 1 ace
(4/52 * 48/51 * 47/50 * 46/49 * 45/48) * 5
possibility of picking ace and 4 non ace cards then * 5 to account for all locations of ace
What is the probability of five card poker hand containing two different pairs and a 5th card of a different type
C(13,2)C(4,2)C(4,2)*C(44,1) / C(52/2)
choose 2 types * pick2 of the 4 for both types, then choose extra card then divide by possible hands
What is the probability of a five card poker hand contains a straight
(C(10,1) * C(4,1)^5) / C(52,5)
choose starting suit, then choose 1 of each necessary cards out of 4 suits, then divide by all possible cards
Find the probability that a randomly generated bit string of length 10 does not contain any 0 bits
c(10,0) * ((1/2)^10)
prove if E and F are independent are !E and F independent
F=SnF=Fn(Eu!E) = (FnE)u(Fn!E) p(F) = p(FnE) + p(Fn!E) p(F) - p(FnE) = p(Fn!E) p(F) - p(F)p(E) = p(Fn!E) p(F)(1-p(E) = p(Fn!E) P(F)p(!E) = p(Fn!E)
Expected number of hads when fair coin flipped 10 times
1/2 * 10 = 5
Expected number of heads when coins is biased and heads comes up .6
.6*10 = 6
Suppose a fair coin is flipped until it either comes up tails twice or is flipped 6 times, what is the expected number of flips
2p(N=2) + 3p(N=3) + 4p(N+4) + 5p(N=5) + 6*(N=6)
3.75
Planar Graph
No edges cross
Euler’s Formula
regions = edges - vertices + 2
If a connected planar simple graph with v>=3
e<= 3v - 6
If connected planar simple graph v>=3 and no circuits of length 3
e <=2v-4
What cannot exist in a planar graph
a subgraph of K3,3 or K5
How many different ways are there to arrange the letters of the word Thanksgiving
12!/(2!2!2!)
How many different ways are there to arrange the letters of the word Thanksgiving that start with T
11!/(2!2!2!)
How many different ways are there to arrange the letters of the word Thanksgiving that contain the word HAT
10!/(2!2!2!)
How many different ways are there to arrange the letters of the word Thanksgiving where the two Gs are next to each other
12!/(2!2!2!) - 11!(2!2!)
all possible ways - ways where GG is next to each other
How many different ways are there to arrange the letters of the word Thanksgiving where vowels are in alphabetical order
9!/(2!2!) * C(12,3)
How many different functions are there from a 5 element set to a 6 element set
6^5
How many different functions are there from a 5 element set to a 6 element set are 1-1
65432
How many different functions are there from a 5 element set to a 6 element set are onto
Impossible since 5 elements in first set and 6 in second
How many different relations are there from a 5 element set to a 6 element set
2^30
