weeks 1 to 7 info

Question 1

How many ways can 10 math books, 3 history books, 5 biology books, and 2 French books be arranged if books of the same subject must stay together?

Combinatorial Analysis

Answer 1

4!×10!×3!×5!×2!4!×10!×3!×5!×2!

Permutations of grouped objects: Multiply permutations within groups by permutations of the groups.

Question 2

How many distinct permutations are there of the word “BALLOON”?

Permutations with Duplicates

Answer 2

7!2!×2!=12602!×2!7!​=1260

Divide by factorial counts of repeated letters (B, A, L×2, O×2, N).

Question 3

A committee of 3 is formed from 10 people. How many committees are possible?

Combinations

Answer 3

(103)=120(310​)=120

(nr)=n!r!(n−r)!(rn​)=r!(n−r)!n!​.

Question 4

18 children are divided into two teams of 9. How many divisions are possible?

Multinomial Coefficients

Answer 4

18!9!9!=48,6209!9!18!​=48,620

Multinomial coefficient for dividing into equal groups.

Question 5

IfP(A)=0.4P(A)=0.4,P(B)=0.5P(B)=0.5, andP(A∩B)=0.2P(A∩B)=0.2, findP(A∪B)P(A∪B).

Probability Axioms

Answer 5

0.4+0.5−0.2=0.70.4+0.5−0.2=0.7

Inclusion-Exclusion Principle:P(A∪B)=P(A)+P(B)−P(A∩B)P(A∪B)=P(A)+P(B)−P(A∩B).

Question 6

In a communication system withnnantennas (mmdefective), what is the probability the system is functional if no two defective antennas are consecutive?

Conditional Probability

Answer 6

(n−m+1m)(nm)(mn​)(mn−m+1​)​

Combinatorial counting: Place defective antennas with gaps.

Question 7

A medical test is 99% accurate for disease detection (1% prevalence). If you test positive, what is the probability you have the disease?

Bayes’ Theorem

Answer 7

0.99×0.010.99×0.01+0.01×0.99≈50%0.99×0.01+0.01×0.990.99×0.01​≈50%

Bayes’ Formula:P(Disease∣+)=P(+∣D)P(D)P(+)P(Disease∣+)=P(+)P(+∣D)P(D)​.

Question 8

A die is rolled. LetXXbe the outcome. ComputeE[X]E[X].

Expected Value

Answer 8

1+2+3+4+5+66=3.561+2+3+4+5+6​=3.5

E[X]=∑x⋅P(X=x)E[X]=∑x⋅P(X=x).

Question 9

CalculateVar(X)Var(X)for a fair die roll.

Variance

Answer 9

E[X2]−(E[X])2=916−3.52≈2.92E[X2]−(E[X])2=691​−3.52≈2.92

Var(X)=E[X2]−(E[X])2Var(X)=E[X2]−(E[X])2.

Question 10

What is the probability of 6 heads in 9 fair coin tosses?

Binomial Distribution

Answer 10

(96)(0.5)9=84×1512≈0.164(69​)(0.5)9=84×5121​≈0.164

PMF:(nk)pk(1−p)n−k(kn​)pk(1−p)n−k.

Question 11

Calls arrive at 2 per minute. FindP(exactly3callsin2minutes)P(exactly3callsin2minutes).

Poisson Distribution

Answer 11

e−4433!≈0.195e−43!43​≈0.195

Poisson PMF:P(k)=e−λλkk!P(k)=e−λk!λk​(λ=2×2=4λ=2×2=4).

Question 12

A batter with a 30% hit rate. What isP(firsthiton4thtry)P(firsthiton4thtry)?

Geometric Distribution

Answer 12

(0.7)3×0.3=0.1029(0.7)3×0.3=0.1029

Geometric PMF:P(X=k)=(1−p)k−1pP(X=k)=(1−p)k−1p.

Question 13

From 10 balls (4 red, 6 blue), 3 are drawn. FindP(2red)P(2red).

Hypergeometric Distribution

Answer 13

(42)(61)(103)=6×6120=0.3(310​)(24​)(16​)​=1206×6​=0.3

Hypergeometric PMF:(Kk)(N−Kn−k)(Nn)(nN​)(kK​)(n−kN−K​)​.

Question 14

GivenF(x)F(x)with jumps at 0, 1, 3, findP(X=3)P(X=3).

Cumulative Distribution Function (CDF)

Answer 14

F(3)−F(3−)=0.9−0.5=0.4F(3)−F(3−)=0.9−0.5=0.4

P(X=a)=F(a)−lim⁡x→a−F(x)P(X=a)=F(a)−limx→a−​F(x).

Question 15

A system has 5 components, each working with probability0.80.8. FindE[workingcomponents]E[workingcomponents].

Linearity of Expectation

Answer 15

5×0.8=45×0.8=4

E[X1+⋯+Xn]=E[X1]+⋯+E[Xn]E[X1​+⋯+Xn​]=E[X1​]+⋯+E[Xn​].

Question 16

Simplify(E∪F)(E∪Fc)(E∪F)(E∪Fc).

Venn Diagrams

Answer 16

EE

Distributive Law:E∩(F∪Fc)=E∩S=EE∩(F∪Fc)=E∩S=E.