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Counting Principles and Probability Concepts
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1
Question
What is the multiplication principle for counting outcomes?
Answer
Total outcomes = product of outcomes at each stage
2
Question
How many ordered outcomes when drawing a card (52) then a die (6) with replacement?
Answer
52 × 6 = 312
3
Question
How many ordered outcomes when drawing two cards with replacement from a 52-card deck?
Answer
52² = 2704
4
Question
How many ordered outcomes when drawing two cards without replacement from a 52-card deck?
Answer
52 × 51 = 2652
5
Question
What is a permutation of n objects taken r at a time?
Answer
An ordered arrangement of r distinct objects from n
6
Question
What is the formula for nPr (permutations)?
Answer
nPr = n(n−1)…(n−r+1) = n!/(n−r)!
7
Question
What is 0! and why is it defined as 1?
Answer
0! = 1 by convention, to make formulas work nicely
8
Question
How many two-card ordered outcomes from 52 cards with replacement?
Answer
52² = 2704
9
Question
How many two-card ordered outcomes from 52 cards without replacement?
Answer
52 × 51 = 2652
10
Question
What is a combination of n objects taken r at a time?
Answer
An unordered selection of r objects from n
11
Question
What is the binomial coefficient for n choose r?
Answer
C(n, r) = n! / [r!(n−r)!]
12
Question
How is C(n, r) related to permutations nPr?
Answer
C(n,r) = nPr / r!
13
Question
What is the number of combinations of 52 cards taken 2 at a time?
Answer
C(52, 2) = 1326
14
Question
What is a distinguishable permutation with all objects distinct?
Answer
Number of permutations is n! for n objects
15
Question
What is a multinomial coefficient used for?
Answer
Number of distinguishable permutations with repeated types
16
Question
What is the form of the multinomial coefficient for counts n1, n2, …, nk with sum n?
Answer
n! / (n1! n2! ... nk!)
17
Question
How is probability defined via counting outcomes?
Answer
P(A) = N(A) / N(S)
18
Question
In a deck of 52 cards, how many 5-card hands are possible?
Answer
C(52, 5) = 2,598,960
19
Question
If N(S) is total outcomes and N(A) is outcomes for A, what is P(A)?
Answer
P(A) = N(A) / N(S)
20
Question
What does sampling with replacement mean?
Answer
After drawing a item, you return it before next draw
21
Question
What does sampling without replacement mean?
Answer
Drawn items are not returned; counts decrease
22
Question
What is the effect of drawing two cards with replacement on independence?
Answer
Draws are independent
23
Question
What is conditional probability P(A|B)?
Answer
Probability of A given B occurred
24
Question
How is P(A|B) defined in terms of counts?
Answer
P(A|B) = N(A ∩ B) / N(B)
25
Question
What is P(A|B) in terms of S, using N(A∩B), N(B)?
Answer
P(A|B) = [N(A∩B)/N(S)] / [N(B)/N(S)]
26
Question
What is the basic conditional probability formula?
Answer
P(A|B) = P(A ∩ B) / P(B) (P(B) > 0)
27
Question
If B ⊂ A, what is P(A|B)?
Answer
P(A|B) = 1
28
Question
If A ⊂ B, what is P(A|B)?
Answer
P(A|B) = P(A) / P(B)
29
Question
What is the complement rule for conditional probability?
Answer
P(A^c|B) = 1 − P(A|B)
30
Question
What does Bayes's theorem relate?
Answer
P(A|B) to P(B|A) via prior P(A) and P(B)