Proofs
a) With conditional probability, P(A|B), the axioms of probability hold for the event on the left side of the bar. A useful consequence is applying the complement rule to conditional probability. We have that P(A|B) = 1 − P(A|B). Prove this by showing that P(A|B) + P(A|B) = 1 (Hint: just use the definition of conditional probability)
b) If two events A and B are independent, then we know P(A ∩ B) = P(A)P(B). A fact is that if A and B are independent, they so are all combinations of A, B, . . . etc. Show that if events A and B are independent, then P(A ∩ B) = P(A)P(B), and thus A and B are independent. (Hint: P(A ∩ B) = 1 − P(A ∪ B). Then use addition rule and simplify.)
Proofs a) With conditional probability, P(A|B), the axioms of probability hold for the event on the...
Write coherent proofs for the following basic consequences of the axioms of probability theory: a. P(Ac)=1−P(A)usingS=A∪Ac disjoint union; b. 0≤P(A)≤1; c. P(∅)=0
2.28 Using the axioms of probability, prove Bonferroni's inequality: For events A and B, P(AB) 2 P(A) + P(B)-1
2. Suppose A and B are two events. Use the axioms of probability to prove the following (a) P(AnB) 2 P(A) P(B) 1 (b) Show that the probability that one and only one of the events A or B occurs is P(A)+ P(B) -2P(AnB). 3. There are 9 lights labeled 1 to 9, and they are lined up in a row in Boelter Hall. or budget reasons, we are going to turn off 3 of them. For security purposes, we...
The event is said to be repelled by the event B if P(AB) . P (A), and to be attracted by B if P(AIB) > P(A). Show that (a) if B attracts A, then A attracts B, and Bc repels A (Hint: use the definition of conditional probability) (b) If A attracts B, and B attracts C, does A attract C? (Hint: consider when A and C are disjoint sets).
3. Using only the three axioms of probability, prove the Bonferroni inequality: P(AUB P(A) P(B)
Analyze the conditional probability P (B/A), for each scenario given in the first column and thus dlassify them as dependent and independent events under 2 column headings. Scenario 1 :'A, be the event that 70% of the children like chocolate cupcakes and B' be the event that 25% like lemon cupcakes, 30% of children like both. Scenario 2 :B' be the event that 60% of the players are selected for offensive side and 'A' be the event that 40% are...
1. Prove“inclusion-exclusion,”thatP(A∪B)=P(A)+P(B)−P(A∩B). 2. Prove the “unionbound, ”thatP(A1∪A2)≤P(A1)+P(A2). Under what conditions does the equality hold? 3. Provethat, for A1 andA2 disjoint, P(A1∪A2|B)=P(A1|B)+P(A2|B). 4. A and B are independent events with nonzero probability. Prove whether or not A and Bc are independent.
Problem 2. (6 pts) Independence and Conditional Probability (a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8. (b) (2pts) Consider any two random variables X, Y of...
1. Consider a statistical experiment E: (, F,P) and an event A . Note: A EF. a. Use the axioms of probability to show that P(A) 1-P(A). b. Repeat (a) using the definition of the σ-field. 2. Consider a statistical experiment E: (, F,P) in which a fair coin is flipped successively until the same face is observed on successive flips. Let A = {x: x = 3, 4, 5, . . .); that is, A is the event that...
7. If A and B are independent events, then P(A and B) equals a. b. c. P(A) + P(B/A). P(A) x P(B). P(A) +P(B). d. P(A/B) +P(B/A) 8.Which formula represents the probability of the complement of event A? b. 1-P(A) c. P(A d. P(A)-1 9. The simultaneous occurrence of two events is called a. prior probability b. subjective probability c. conditional probability d. joint probability 10. If the probability of an event is 0.3, that means the event has a...