Classification and Partitions
A fair die is rolled. Write the sample space Ω.
What is Ω = {1, 2, 3, 4, 5, 6}?
The end of each complete path on a tree is called this.
What is a leaf?
Ω = {1, 2, 3, 4, 5, 6}. P(E) = 1/2 and P(H | E) = 2/3. Find P(E and H).
What is 1/3 ?
Events A and B are independent when P(A) × P(B) equals this.
What is P(A and B)?
The bar in P(A | B) is read this way.
What is “given that”?
Two events with no outcomes in common are called this.
What is disjoint (mutually exclusive)? (A ∩ B = ∅)
To find the probability of a leaf, you do this to the branch probabilities on its path.
What is multiply?
Ω = {1, 2, 3, 4, 5, 6}. Find P(E and L).
What is 1/6 ?
P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.2. Are A and B independent?
What is yes? (0.5 × 0.4 = 0.20)
P(A | B) equals P(A and B) divided by this.
What is P(B)?
Is {1, 2}, {3, 4}, {5, 6} a partition of Ω for one die?
What is yes? (The pieces don’t overlap and their union is Ω.)
The probabilities of all the leaves on a tree must add to this.
What is 1?
Ω = {1, 2, 3, 4, 5, 6}. Find P(Eᶜ and H).
What is 1/6 ?
A fair die is rolled. A = roll even and B = roll greater than 3. Independent?
What is no? 1/2 * 1/2 = 1/4 but P(A and B) = 1/3
The leaves J ∩ C and Jᶜ ∩ C have probabilities 0.20 and 0.06. Find P(C).
What is 0.26?
Is {1, 2, 3} and {3, 4, 5, 6} a partition of Ω? Explain.
What is no? (They overlap at 3, so they are not disjoint.)
A second-stage branch label such as P(B | A) is this type of probability.
What is conditional?
Ω = {1, 2, 3, 4, 5, 6}. Add the leaves to find P(H).
What is 1/2 ?
A fair die is rolled. A = {1, 2} and B = {1, 3, 5}. Independent?
What is yes? P(A) = 1/3 and P(B) = 1/2, (1/2 * 1/3 = 1/6), and P(A and B) = 1/6
P(J and C) = 0.20 and P(C) = 0.26. Find P(J | C).
What is 10/13 , about 0.77?
Let E = {2, 4, 6} and H = {4, 5, 6}. Write the chain of proper subsets from the leaf E ∩ H up to Ω.
What is {4, 6} ⊂ {2, 4, 6} ⊂ Ω?
Write the leaf for the path J then C in set notation, and give its subset relationship to J.
What is J ∩ C, with (J ∩ C) ⊂ J?
A bag has 3 red and 2 blue marbles. Two are drawn without replacement. Find P(R₁ and R₂)
What is 3/10 ?
P(A) = 0.6, P(B) = 0.3, and P(A or B) = 0.72. Independent?
What is yes? (P(A and B) = 0.6 + 0.3 − 0.72 = 0.18 = 0.6 × 0.3)
2% of phones are defective (D). An inspector flags (F) 90% of defective phones and falsely flags 5% of good ones. Find P(D | F).
What is 18/67, about 0.27? (P(D and F) = 0.018 and P(Dᶜ and F) = 0.049, so P(F) = 0.067.)