Probability Test #1
Intersection means
"AND"
Union means
"OR"
Equally likely outcome formula
#(A)/#(Ω)
At least two of the three events (A,B,C) occur
(A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C)
Either C and B occur or A does not occur
(B ∩ C) ∪ A^c
For any event A, P(A) >=
0
Probability of the sample space S is P(S) =
1
For any event A, P(A) <=
1 because from up above, P(A^c) >=0
For any event A, P(A^c) =
1 - P(A)
You roll a fair die. What is the probability of E={1,5} ?
1. The problem states that the die is fair, which means that all six possible outcomes are equally likely 2. In particular, since the events {1},{2},⋯,{6} are disjoint we can write 1 = P( {1} ∪ {2} ∪ ... {6}) = P(1) + P(2) + ... + P(6) 3. P({1}) = P({2}) =P({6})= 1/6 4. Again since {1} and {5}are disjoint, we have P(E) = P({1,5}) = P(1) + P(5) = 2/6 = 1/3
Outcome
A result of a random experiement
A and B occur, but not C
A ∩ B ∩ C^c
Ω
All possible outcomes
a) Chance for first try then second try with replacement means that: b) what's the combined probability?
Both tries have the same probability because of replacement If you want a combined probability, multiply the 2 probabilities together
Event
Collection of possible outcomes or a subset of the sample space to which we assign a probability
Dependent Events
Dependent events are events where an outcome of one event IS affected by the outcome of the other event
A box contains tickets marked 1,2,...,n. A ticket is drawn at random from the box. Then this ticket is replaced in the box and a second ticket is drawn at random. Find the probabilities of the following events: b) The numbers on the two tickets are consecutive integers, meaning the first number drawn is one less than the second number drawn
First pick: 1/n - Pick any card Second pick: n-1/n -Second pick cant be the first card so its n-1 1/n x (n-1)/n = (n-1)/n^2
Trial
If we repeat a random experiement several times
Independent Events
Independant events are events where an outcome of one event is NOT affected by the outcome of the other event
A∩B
Intersection of A and B -Set of outcomes where both A ~and~ B happen at once
"and" "all of" correspond to
Intersections
At most n of the k events occur
Make all combos "and" and "complementary"
At Least n of the k events occur
Make all combos "intersect"
If events A and B are independent, then the probability of both A and B occurring is
P(A and B) = P(A) x P(B)
If Events A and B are dependent, then the probability of both A and B occurring is
P(A and B) = P(A) x P(B|A)
P(A ∩ B)
P(A and B) or P(AB)
P(A ∪ B) =
P(A or B)
P(A) =
P(A ∩ B) + P(A ∩ B^c)
Inclusion-exclusion Principle for ABC
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B ∩ C)
Inclusion-exclusion Principle
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A ∪ B) =
P(A) + P(B) - P(A ∩ B)
P(A - B) =
P(A) - P(A ∩ B) *Think of a ven diagram
In summary if A1 and A2 are disjoin events, then P(A1 ∪ A2) =
P(A1) + P(A2) and the same thing is true when you have n disjoint events A1, A2, ..., An.
If A1,A2,A3,... are disjoint events, then P(A1 ∪ A2 ∪ A3 ...) =
P(A1)+P(A2)+P(A3)+...
P[(A ∪ B)^c] =
P(A^c ∩ B^c)
The probability of the empty set is
P(∅) = 0
Sample space of 1. Rolling a die 2. Tossing a coin 3. # of iPhone sold by an apple store in Boston in 2015
S = {1,2,3,4,5,6} S = {heads, tails} S = {1,2,3,4,....}
Write the sample space S for the following random experiments. a) We toss a coin until we see two consecutive tails. We record the total number of coin tosses.
S = {2,3,4,....}
b) A bag contains 4 balls: one is red, one is blue, one is white, and one is green. We choose two distinct balls and record their color in order.
S = {R, B}, {R, W}, {R, G}, {B,W}, {B,G}, {W,G}
Sample Space
The set of all possible outcomes
A∪B
The union of A and B -All outcomes in A or in b or in both
In a presidential election, there are four candidates. Call them A, B, C, and D. Based on our polling analysis, we estimate that A has a 2020 percent chance of winning the election, while B has a 4040 percent chance of winning. What is the probability that A or B win the election?
This is a union of A and B P(A) + P(B) = .40+.20 = 0.60
"or" "at least" correspond to
Unions
At most one of the three events occurs.
[(A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C)]^c
A box contains tickets marked 1,2,...,n. A ticket is drawn at random from the box. Then this ticket is replaced in the box and a second ticket is drawn at random. Find the probabilities of the following events: a) the first ticket is drawn is number 1 and the second ticket drawn is number 2
a) 1/n x 1/n = 1/n^2
