Probability and Combinatorics Flashcards
The
probability of flipping a
fair coin once and getting
heads can be expressed as
1
2
Respectively, the
numerator and
denominator of this fraction
can be thought of as what?
**_# of possibilities that meet conditions_ # of equally likely possibilities**
# _define_: **trial** and **experiment**
Trial:
one
individual performance of a
random experiment
Experiment:
any procedure that can be
performed infinitely and has a
well-defined set of
possible outcomes
(basically a set of a number of trials)
# _define_: sample space
The
- *set** of
- *all possible outcomes**
- e.g.*
- In a six-sided di**, {1, 2, 3, 4, 5, 6}*
What is the
name and
meaning of the
symbols below?
P(A)
name:
Probability function
meaning:
Probability of event A
What is the
name and
meaning of the
symbols below?
P(A U B)
name:
Probability of events union
meaning:
Probability that of events A or B
What is the
name and
meaning of the
symbols below?
P(A ∩ B)
name:
Probability of events intersection
meaning:
Probability that of events A and B
What is the
name and
meaning of the
symbols below?
P(A | B)
name:
Conditional probability function
meaning:
Probability of event A given that B occurred
If A and B are
mutually exclusive events,
then how would you
calculate
P(A U B) = _____?
P(A U B) = P(A) + P(B) − P(A ∩ B)
Add the
probabilities of each of them occurring
and
subtract the probability of them both occurring.
Subtract the overlap.
What is the
addition rule for probability?
P(A U B) = P(A) + P(B) − P(A ∩ B)
For mutually exclusive events, add the probabilities of each of them occurring and subtract the probability of both occurring
- e.g.*
- To get to school, Harry can take the train, a car, or a broomstick. He has the same options when he leaves school. If he chooses his mode of transportation randomly to and from school, what is the probability that he takes a car at least once?*
P(C U C) = 1 + 1 − 1 = 5
3 3 9 9
An average
deck of playing cards has
4 suits (H, C, S, D) and
13 ranks (2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A),
totaling
52 cards.
If you
draw one card,
what is the probability that it is a
spades or a Jack?
P(S U J) = 16 = 4
52 13
P(S U J) = P(S) + P(J) − P(S ∩ J)
= 13 + 4 − 1
52 52 52
= 16
52
= 4
13
# _define_: independent event
One event
- *in no way affects**
- *what happens** in
- *other events**
If you
- *add** the
- *probabilities** of
- *all possible**
- *independent events**, what do you get?
1
- Only independent events*
- e.g.*
- The probabilities of randomly picking each of the four cardinal directions is ¼.*
- Added together, this is ¼ + ¼ + ¼ + ¼ = 1*
In craps, with
two fair six-sided dice,
how do you
calculate the probability of rolling
snake eyes?
- P(A* ∩ B) = P(A) • P(B),
- (given independent events A and B)*
- P(A* ∩ B) = P(A) • P(B)
- P(11* ∩ 12) = P(11) • P(12)
= 1/6 • 1/6
= 1/36
On a multiple-choice exam question, there is
one correct answer, and the
probability of randomly selecting that answer is
x/y.
What is the
probability of randomly selecting an
incorrect answer?
1 − x/y
How do you
- *calculate** the
- *probability** that a
- *fair coin** will land on
- *heads**
- *92 times** in a row?
P(H)92
These are independent events, so you can
multiply the probabilities.
- P(H*) = ½
- P(H*)7 = (½)92
= 1
4,951,760,157,141,521,099,596,496,896
≈ 2.01948 x 10−28
If you
randomly select
one of the 50 states,
50 times,
how can you
calculate the probability that you will select
Texas
at least once?
1 − (49/50)50
P(Texas at least once)
= P(not only other states in 50 trials)
= 1 − P(only other states in 50 trials)
= 1 − (49/50)50
≈ 0.63583
William Shakespeare wrote
- *37 plays**, including
- Henry IV, Part 1* and
- Henry IV*, Part 2.
If you
randomly select two different Shakespearean plays to read,
how would you calculate the probability that you select those two plays
in that order?
P(A ∩ B) = P(A) • P(B | A)
Given
dependent events A & B,
t**he probability that A and B will occur is
equal to the
probability of A times the
probability of B given that A occurred.
P(1stPart 1 ∩ 2ndPart 2) = P(1stPart 1) • P(2ndPart 2 | 1stPart 1)
= 1 • 1
37 36
= 1
1,332
Given fair dice,
how would you
calculate the
probability of rolling
two natural 20s on a d20 followed by
two 8s on a d8?
(1/20)2 • (1/8)2 = 1/256
These are independent events, so the respective probabilities can be multiplied
Mathematically,
what does
n!
mean?
n factorial
or
n times
every natural number
less than n
or
n • (n − 1) • (n − 2) • … • 1
What does
0!
mean?
1
0 factorial is
defined to be 1 so that the
formulas work
# _define_: permutation
An
- *arrangement** of
- *items** in a
- *particular order**
e.g.
ABCD and ABDC are
two permutations
# _define_: combination
An
- *arrangement** of
- *items** in which
- *order is irrelevant**
e.g.
ABCD and ABDC are
one combination
What is the
permutations formula?
nPk = n!
(n − k)!
What is the
combinations formula?
nCk = n!
k! • (n − k)!
