Place your order now for a similar assignment and have exceptional work written by our team of experts, At affordable rates

H6. • The assignment is due at Gradescope on Monday, October 17 at 10pm. Submit early and often.
• Read and sign the collaboration and honesty policy. Submit the signed policy to Gradescope before submitting any work.
• Unless otherwise specified, you can leave your answer in closed form (e.g. 1 − (120)(0.1)200). 7
• Show your work. Answers without justification will be given little credit. Justify each step in your solutions e.g. by stating that the step follows from an axiom of probability, a definition, algebra, etc.; for example, your answer could include a line like this:
Pr(X ∩ Y ∩ Z) · Pr(A ∪ B) ＝ Pr(X ∩ Y ∩ Z) · (Pr(A) + Pr(B)) (A and B are disjoint)
• The syllabus has some pointers on using LaTeX and Python.
problem 1. Consider the following experiment. We start with a standard deck of 52 cards. We repeatedly draw a card from the deck without replacement until we draw an Ace and then draw one additional card, also without replacement. (For example, we might draw K♡, 2♢, A♠, 3♣.) In each draw, the card is chosen uniformly at random from the remaining cards.
We define the following random variables:
N ＝ total number of cards that we draw
D ＝ total number of diamond cards that we draw A ＝ total number of Ace cards that we draw
(a) Identify the sample space for this experiment.
(b) Consider the outcome ω corresponding to drawing K♡, 2♢, A♠, 3♣. Find N(ω), D(ω), A(ω).
(c) Find the range of N, D, and A.
(d) Find the probability that we draw exactly 3 cards, i.e., Pr(N ＝ 3).
(e) Find the probability that the last card is not an Ace given that we draw 3 cards in total, i.e., Pr(”last card is not an Ace”|N ＝ 3).
problem 2. (a) We toss a fair coin three times. Find the probability that exactly two heads occur, given
that the first toss was a heads.
(b) We roll a standard 6-sided die twice. Find the probability that the sum of the faces is greater than 7,
given that the first roll was less than 5.
(c) We roll two standard 6-sided dice once. Find the probability that the sum of the two rolls is 6 given
that the dice land on different numbers.
(d) Let x be a point selected uniformly at random from the interval [0,1]. Find the probability that x ˃ 1/2, given that x2 − x + 2/9 ˂ 0.
(e) We toss a dart at a circular target of radius 4 inches. Given that the dart lands in the upper half of the target, find the probability that its distance from the center is greater than 2 inches.
problem 3. Suppose you are playing a round of roulette at a casino, in which the board consists of {1, 3, 5, 7, 9} as black numbers, {2, 4, 6, 8, 10} as red numbers, and 0 as a green number. You can choose to bet on red, black, or a specific number besides 0 (you can make multiple bets in a single round). A wheel and a ball is used to randomly uniformly select one of the numbers, and if it lands on your specific number or any number in your chosen color (if you bet red/black) you will win. Note that the green 0 is always a loss.
(a) If you bet on 10 and also bet on black in a single round, what is your probability of winning?
(b) Given that the ball will select a black number, what is your probability of winning if you bet on 1 and red?
(c) Given that the ball will select a number less than or equal to 7, what is your probability of winning if you bet on red?
problem 4. Amos Tversky1 and Daniel Kahneman2 are behavioral psychologists that are well-known for their empirical studies on how people’s intuitions about probability are at odds with mathematical facts. In one of their experiments, they considered the following scenario:
Let A be the event that before the end of next year, Peter would have installed a burglar alarm system in his home. Let B be the event that Peter’s home will be burglarized by the end of next year.
(a) Intuitively, which do you think it is bigger: Pr(A|B) or Pr(A|B)? Explain your intuition.
(b) Intuitively, which do you think it is bigger: Pr(B|A) or Pr(B|A)? Explain your intuition.
(c) Show that, for any two events A and B (with probabilities not equal to 0 or 1), the inequality Pr(A|B) ˃ Pr(A|B) is equivalent to Pr(B|A) ˃ Pr(B|A).
Hint: Start by drawing the Venn diagram showing A, A, B, B and their pairwise intersections. Ex- press the probabilities of interest in terms of the probabilities of the events A ∩ B, A ∩ B, A ∩ B, A ∩ B. To simplify your notation, you can name each of these probabilities, e.g., as p, q, r, s.
(d) In Tversky and Kahneman’s experiment, 131 out of 162 people said that Pr(A|B) ˃ Pr(A|B) and Pr(B|A) ˂ Pr(B|A). What is a plausible explanation for why this was such a popular response, despite (c) showing that this is mathematically impossible?