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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).

Solution: Your solution here.

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.

Solution: Your solution here.

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?

Solution: Your solution here.

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?

Solution: Your solution here.

problem 5 (Programming exercises). Download the HW6 Jupyter notebook from Piazza. Complete all the exercises in the notebook. Submit the Jupyter notebook with your solutions to the Homework 6 Programming assignment on Gradescope. Your submission should be a single .ipynb file.

1 https://en.wikipedia.org/wiki/Amos_Tversky

2 https://en.wikipedia.org/wiki/Daniel_Kahneman

H6-2

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