Probability

Question 1

p(E) = ?

Answer 1

|E|/|S|

Question 2

p(E | F)

Answer 2

p(E n F) / p(F)

Question 3

P(!E) = ?

Answer 3

1 - p(E)

Question 4

Independence of E and F

Answer 4

p(E n F) = p(E) * p(F)

Question 5

Independence of p(E | F) = ?

Answer 5

p(E)

Question 6

Bays Thereon p(F | E) = ?

Answer 6

( p(E | F) * p(F) ) / ( ( p(E | F) * p(F) + p(E | !F) * p(!F) )

Question 7

Probability of k successes out of n trials
p = probability of success
q = probability of failure

Answer 7

C(n,k)(p^k)(q^n-k)

Question 8

p(E1 u E2)

Answer 8

p(E1) + p(E2) - p(E1 n E2)

Question 9

probability 5 card poker hand has ace of hearts

Answer 9

1/52 + 1/52 + 1/52 + 1/52 + 1/52 = 5/52

Question 10

probability 5 card poker hand does not have queen of hearts

Answer 10

1 - (1/52 + 1/52 + 1/52 + 1/52 + 1/52) = 47/52

Question 11

probability 5 card poker hand contains 2D, 3S, 6H 10C, KH

Answer 11

1/C(52,5)

Question 12

probability that five card poker hand contains at least 1 ace

Answer 12

1-(C(48,5)/C(52,5))

Question 13

probability that a five card hand contains exactly 1 ace

Answer 13

(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

Question 14

What is the probability of five card poker hand containing two different pairs and a 5th card of a different type

Answer 14

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

Question 15

What is the probability of a five card poker hand contains a straight

Answer 15

(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

Question 16

Find the probability that a randomly generated bit string of length 10 does not contain any 0 bits

Answer 16

c(10,0) * ((1/2)^10)

Question 17

prove if E and F are independent are !E and F independent

Answer 17

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)

Question 18

Expected number of hads when fair coin flipped 10 times

Answer 18

1/2 * 10 = 5

Question 19

Expected number of heads when coins is biased and heads comes up .6

Answer 19

.6*10 = 6

Question 20

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

Answer 20

2p(N=2) + 3p(N=3) + 4p(N+4) + 5p(N=5) + 6*(N=6)

3.75

Question 21

Planar Graph

Answer 21

No edges cross

Question 22

Euler’s Formula

Answer 22

regions = edges - vertices + 2

Question 23

If a connected planar simple graph with v>=3

Answer 23

e<= 3v - 6

Question 24

If connected planar simple graph v>=3 and no circuits of length 3

Answer 24

e <=2v-4

Question 25

What cannot exist in a planar graph

Answer 25

a subgraph of K3,3 or K5

Question 26

How many different ways are there to arrange the letters of the word Thanksgiving

Answer 26

12!/(2!2!2!)

Question 27

How many different ways are there to arrange the letters of the word Thanksgiving that start with T

Answer 27

11!/(2!2!2!)

Question 28

How many different ways are there to arrange the letters of the word Thanksgiving that contain the word HAT

Answer 28

10!/(2!2!2!)

Question 29

How many different ways are there to arrange the letters of the word Thanksgiving where the two Gs are next to each other

Answer 29

12!/(2!2!2!) - 11!(2!2!)

all possible ways - ways where GG is next to each other

Question 30

How many different ways are there to arrange the letters of the word Thanksgiving where vowels are in alphabetical order

Answer 30

9!/(2!2!) * C(12,3)

Question 31

How many different functions are there from a 5 element set to a 6 element set

Answer 31

6^5

Question 32

How many different functions are there from a 5 element set to a 6 element set are 1-1

Answer 32

65432

Question 33

How many different functions are there from a 5 element set to a 6 element set are onto

Answer 33

Impossible since 5 elements in first set and 6 in second

Question 34

How many different relations are there from a 5 element set to a 6 element set

Answer 34

2^30