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exam_i
- Combinatorics including
- Dice
- Card Hands
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- Binomial Expansion
- Geometric Series
- Exponential Series
- Logrithmic Series
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- Binomial coefficients
- Multinomial coefficients
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- Axioms of probability
- Conditional probability
- Baye's formula
- Independence
- Inclusion-exclusion
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- Expectation
- Variance
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- Binomial random variable [(n,p)]
- Poisson random variable [(\lambda)]
- Geometric random variable [(p)]
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- Stirling's Formula
- Integrals of probability theory
[{n \choose i} = \frac{ n!}{ (n - i)! i!}]
where [0 \le i \le n]
[(x + y)^n = \sum\limits_{i = 0}^{n}{ {n \choose i} x^i y^{n-i}}]
For nonnegative integers [n_1, \dots, n_r] summing to [n]
[{n \choose n_1, n_2, \dots, n_r} = \frac{ n!}{ n_1! n_2! \cdots n_r!}]
is the number of division of [n] items into [r] distinct non-overlapping subgroups of sizes [n_1, n_2, \dots, n_r]
- [0 \le P( A) \le 1]
- [P( S) = 1]
- For mutually exclusive events [A_i, i \ge 1]
[P\left(\bigcup\limits_{i = 1}^{\infty}{ A_i}\right) = \sum\limits_{i = 1}^{\infty}{ P( A_i)}]
[P(A \cup B) = P( A) + P( B) - P( AB)]
[E] given [F] has occurred is denoted
[P( E|F) = \frac{ P( EF)}{ P( F)}]
[P( E_1 E_2 \cdots E_n) = P( E_1) P( E_2|E_1) \cdots P( E_n|E_1 \cdots E_{n - 1})]
[P( E) = P( E|F)P(F) + P( E|F^c)P( F^c)]
[P( F_j | E ) = \frac{ P( E|F_j)P( F_j)}{ \sum\limits_{i = 1}^{n}{ P( E|F_i) P( F_i)}]
[E[ X] = \sum\limits_{x:p( x) > 0}{ x p( x) }]
[Var( X) = E[ (X - E[ X])^2] = E[ X^2] - (E[X])^2]
pmf [p( i) = {n \choose i} p^i (1 - p)^{n - i}]
[E[ X] = n p]
[Var( X) = np(1 - p)]
pmf [p( i) = \frac{ e^{-\lambda} \lambda^i}{ i!}, i \ge 0]
[E[ X] = \lambda]
[Var( X) = \lambda]
pmf [p( i) = p(1 - p)^{i - 1}, i = 1, 2, \dots]
[E[ X] = \frac{ 1}{ p}]
[Var( X) = \frac{ 1 - p}{ p^2}]
[k! \sim k^{k + \frac{ 1}{ 2}} e^{-k} \sqrt{ 2 \pi}]
[20] workers are to be assigned to [20] different jobs, one to each job. How many different assignments are possible?
- [20!]
- [20] unique jobs and [20] unique workers
In how many ways can 3 boys and 3 girls sit in a row?
- [6!]
- [6] unique individuals
In how many ways can [3] boys and [3] girls sit in a row if the boys and the girls are each to sit together?
- [(2) 3! 3!]
- [2] groups of [3] unique individuals filling [3] seats per group with [2] possible arrangements of the groups
In how many ways if only the boys must sit together?
- [(4) 3! 3!]
- [2] groups of [3] unique individuals filling [3] seats per group with [4] possible arrangements of the groups as the groups, the group of boys having four positions between the girls.
In how many ways if no two people of the same sex are allowed to sit together?
- [(2) 3! 3!]
- [2] groups of [3] unique individuals filling [3] seats per group with [2] possible arrangements of the groups
How many different letter arrangements can be made from the letters
- Fluke
- [5!]
- All unique letters
- [5!]
- Propose
- [\frac{ 7!}{ 2! 2!} ]
- [7] letters, [5] unique, [2] sets of [2] same latter
- [\frac{ 7!}{ 2! 2!} ]
How many 5-card poker hands are there?
- [{52 \choose 5}]
Two dice are thrown. Let [E] be the event that the sum of the dice is odd, Let [F] be the event that at least one of the dice lands on [1], Let [G] be the event that the sum is [5].
Describe the events.
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[E F]
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[E \cup F]
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[F G]
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[E F^c]
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[E F G].
Suppose that [A] and [B] are mutually exclusive events for which [P( A) = .3] and [P( B) = .5].
What is the probability that
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either [A] or [B] occurs?
- [P( A \cup B) = .8]
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[A] occurs but [B] does not?
- [P( A \cap B^c) = P( A)]
Poker dice is played by simultaneously rolling 5 dice.
- [P{\text{two pair}}]
- [\frac{ {6 \choose 2} {5 \choose 2} {3 \choose 2} {4 \choose 1}}{ 6^5}]
A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability that a 5 occurs first.
[P( 5) = { (1,4), (2,3), (3,2), (4,1)} = \frac{ 4}{ 36} = \frac{ 1}{ 9}] [P( 7) = { (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} = \frac{ 6}{ 36} = \frac{ 1}{ 6}] [P( (7 \cup 5)^c) = \frac{ 13}{ 18}] Let [E] be the even that a 5 occurs before a 7. [P(E_n) = \left(\frac{ 13}{ 18}\right)^{n-1} \frac{ 1}{ 9} = ]
A total of 46 percent of the voters in a certain city classify themselves as Independents, whereas 30 percent classify themselves as Liberals and 24 percent say that they are Conservatives. In a recent local election, 35 percent of the Independents, 62 percent of the Liberals, and 58 percent of the Conservatives voted. A voter is chosen at random. Given that this person voted in the local election, what is the probability that he or she is
Let [L] the event Let [V] be the event this person voted
[P( V) =