SIG Prep - 3 [PDF]

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Technical questions • 27 * 29. 27*30 – 27 = 810 – 27 = 783. • 19 * 19. 361, 20 * 20 – 20 – 19 = 400 – 39 = 361. • 17 * 3, then *3, then *3, then *3, then *3. 17 * 3 = 51. 51 * 3 = 153. 153 * 3 = 459. 459 * 3 = 450 * 3 + 27 = 1377. 1377 * 3 = 3900 + 3 * 77 = 3900 + 210 + 21 = 4131. • 79^2. 80 * 80 – 80 – 79 = 6400 – 159 = 6241. • 39^2. 40 * 40 – 40 – 39 = 1600 – 79 = 1521. • 29^2. 30 * 30 – 30 – 29 = 841. • 19 * 21. (20 – 1)(20 + 1) = 400 – 1 = 399. • 17 * 13. (15 + 2)(15 – 2) = 225 – 4 = 221. • 27 * 33. (30 – 3)(30 + 3) = 900 – 9 = 891. • What is 92 divided by 9? 10 + 2/9 • What is the sum of 1-40? 41 * 20 = 820. • What is the square root of 2000? sqrt(2000) = 10 * sqrt(20) which is slightly less than 45. • EV of a dice? 3.5. • What’s bigger, epi or pie? epi, for numbers greater than 2, bigger exponent matters more. • If you have a 6-sided cube that has a unique color on each side (6 different colors), how many possible combinations of unique cubes are there? 3 pairs of opposite colors, first pair is [6 choose 2] = 15 ways, second pair is [4 choose 2] = 6 ways, third pair is 1 way. Order of pairs don’t matter so divide by 3! = 15 * 6 / 6 = 15 ways. Each pair can be rearranged so 15 * 2 = 30 ways. • What is the chance of drawing two cards from the top of the deck that turns out to be the same number? 1 * 3/51. • EQUATION FOR CLOCK PROBLEMS: angle = | 0.5(60H – 11M) | • What is the angle between the hour and minute hand at 3:30? 30 degrees per hour for the hour hand, so 30 + 30 + 15 = 75 degrees. • What is the angle between the hour and minute hand at 3:15? 30 degrees per hour for the hour hand, so 15/60 * 30 = 7.5 degrees. • What is the angle between the hour and minute hand at 5:12? 12/60 is 1/5, so hour hand is 6 degrees past 5pm. Each minute is 6 degrees. So 5 * 30 + 6 – 72 = 84 degrees. • What is the sum of the digits of your phone number? Square it. Then divide it by 5. 304-5534973. 7 + 10 + 3 = 20, 20 + 10 + 13 = 43. 43 * 43 = 40 * 43 + 129 = 1849. 1849 / 5 = 369.8. • What is the expected value of a deck of cards? Ignore suits. Average of 1 and 13 is 7. • How many 5-card poker hands are there that make 4 of a kind? 13 different 4 of a kinds, 5th card can be one of 48 other cards. 13 * 48 = 650 – 26 = 624. • What is a Markov Chain? A stochastic system in which the current state only depends on the one previous state. Also known as an autoregressive-1 model. • What is the chance that at least two people were born on the same day of the week if there are 3 people in the room? 1 – chance that nobody is born on the same day of the week = 1 – (1)(6/7)(5/7) = 19/49 • Two ropes are non-uniform in composition. They burn completely in 1 hour. You only have a lighter and those two ropes and you have to tell when 45 minutes have passed. Light both ends of one rope, and one end of the other rope. After the first rope burns out, 30 minutes will have passed, and light the other end of the second rope. When the second rope burns out, 15 more minutes will have passed, totaling 45 minutes. • I have a jar filled with 100 pennies. 1 of the pennies has two heads, and the others are normal pennies. I randomly choose one penny, flip it 7 times, and get 7 heads. Am I holding

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the two-headed penny or a normal penny? Chance of the two-headed penny: 1/100. Chance of getting a normal penny and flipping seven heads 99/100 * 1/128. You have a higher chance of having the two-headed penny. I am thinking of a number that does not contain the digit 1. The product of the digits is 96. What is the smallest possible number in my mind? How about the largest? 96 is 2^5 * 3. Largest is 322222. You can group three 2’s together and another 2 and 3 together into 8 and 6, respectively. There is one last 2 left, so you make the smallest number with 8, 6, and 2, which is 268. Player 1 chooses a number, then player 2 chooses a different one. 2 dice are tossed, and whoever is closer to the sum wins. Would you be P1 or P2? What number do you choose? Be player 1, since the best choice is the number 7. You roll a die, and I give you that number in dollars. You can accept at any time but have a max of 3 rolls and must accept the third. How do you maximize your return? Expected value of 3rd roll is 3.5. So pick second die if 4, 5, or 6. Expected value of 2nd and 3rd rolls is ((4 + 5 + 6)/3 + 3.5)/2 = 4.25, so pick 5 or 6 for 1st roll. What is the expected value to you of the following game? You and another person have 3 coins each. You both flip all of your coins and if your three coins show the same number of tails as the other person, you pay the $2, otherwise they pay you $1. Probability that you lose is (1/8)^2 + (1/8)^2 + (3/8)^2 + (3/8)^2 = 20/64 = 5/16. (-2)*(5/16) + (11/16) = 1/16. You have 2/3 chance of hitting a target. What is the chance of hitting it if you fire twice? 1 – no hits = 1 – (1/3)(1/3) = 8/9. You see that the target has been hit. What is the chance that you hit it twice? Probability both hit / probability you hit at all = (4/9) / (8/9) = 1/2. There are 99 lions and 1 sheep on an island. Lions want to eat sheep but also want to stay alive. When a lion eats a sheep, it turns into a sheep. How many lions and sheep left are left after a period of time? (Lions can survive on other foods on the island, sheep cannot escape lions, and all creatures are rational) If there were 2 lions, and 1 sheep, the lions wouldn’t eat because the other lion would eat the transformed lion. However if there were 3 lions, the 3rd lion would eat the sheep knowing the previous fact. Continue and notice that odd lions eat and even ones don’t. So the long-term expected numbers of lions and sheep are 98 and 1, respectively. You roll two dice. What is the probability of rolling a “3” total before rolling a “7” total? Probability of 3 is 1/18. Probability of 7 is 1/6. 1 – 1/18 – 1/6 = 7/9. P = (1/18) + (7/9)P, P = 1/4. What is the probability that when you arrange 5 people in a round table, they are in increasing age order (clockwise or anti-clockwise)? 2 possible arrangements (increasing or decreasing), 5 starting positions, 5! combinations. 2*5/5! = 10/120 = 1/12. If a 5x5x5 Rubik’s cube is sitting on a desk, exactly how many cubes have only one side touching the air? 9 * 5 = 45. If you have a deck of cards split into 4 piles and was offered 1:1 odds to draw a face card (J Q K A) from at least one of the piles, would you take the game? Why or why not? Don’t draw a card from any pile = 1 – (36/52)(35/51)(34/50)(33/49) . If it’s greater than 0.5, take the game. If the chance of you seeing a shooting star outside within one hour is 84 percent, what is the chance of you seeing at least one shooting star outside in 30 minutes? Shooting stars follow a Poisson distribution with parameter lambda, and not seeing a shooting star in one hour has probability 16%. 0.16 = exp(-lambda), lambda / 2 (because of half an hour) = - ln (0.4) = 0.92. So 1 - exp(-0.92) = 60% chance of seeing a shooting star in 30 minutes.

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If I roll a dice and pay you the amount showing, how much would you pay to play? EV of dice = $3.50 If you are given the choice to re-roll after the initial roll, making the initial result null, and you are paid based on the second roll. How much would you pay to play? First roll is worth $3.50, so you would only roll second if first roll was not a 4, 5, or 6. So the second roll is worth $5. 0.5 * 3.5 + 0.5 * 5 = $4.25 Three rolls? First roll is worth $3.50. Second roll is worth $5. First two rolls are worth $4.25, so you would only keep first roll if you get a 5 or a 6. 1/3 * 5.5 + 1/3 * 5 + 1/3 * 3.5 = $4.67 Given a chart with different poker tournaments with different entry fees, reward pots, and probabilities to win, explain how you would gauge the risk against potential profits. Given a bunch of balls of different weights and a scale, and all the balls look identical but one weighs different, give the minimum number of tries with the scale to find the heavy ball. Weight half at a time. Answer is the power of 2 that the number of balls is closest to. A family has 2 kids. If you have seen 1 girl, what is the probability that the other kid is a boy? After seeing one girl, the possible combinations are girl-boy, boy-girl, and girl-girl. Therefore the probability that the other kid is a boy is 2/3. If you flip a coin until you decide to stop and you want to maximize the ratio of heads to total flips, what is that expected ratio? 50% chance of 1:1 ratio initially (flip heads first), because then you would stop. If you start with a tail, you would keep flipping until you have a 1:2 ratio (50% heads). Therefore your expected ratio is 3:4. Find the heavier ball of the given 12 pool balls using a weight balance that can be used just 3 times. Weight 6 vs. 6, then 3 vs. 3, then 1 vs. 1. Two people take turns counting to 50. Each person can add from 1 to 10 to the total sum. The first to count the number 50 wins. What is your strategy? Do you want to go first? You want to start first and hit “checkpoint” numbers 6, 17, 28, and 39.

You have five pirates, ranked from 5 to 1 in descending order. The top pirate has the right to propose how 100 gold coins should be divided among them .But the others get to vote on his plan, and if fewer than half agree with him, he gets killed. How will the coins end up being divided, assuming all the pirates are rational and want to end up alive? You have 8 marbles and they all are identical looking in size, color, and shape. One marble weighs more than the other 7. You have a balance and can use it twice. How would you find out which marble is the heavy one? Weight 3 and 3. If the same, then weight the other two to find heavier one. If different, take 2 out of the heavier 3 and weight them to find the heavier one.

2-dice sum probabilities: 2: 1 3: 2/36 = 1/18 4: 3/36 = 1/12 5: 4/36 = 1/9 6: 5/36 7: 6/36 = 1/6 8: 5/36

9: 4/36 = 1/9 10: 3/36 = 1/12 11: 2/36 = 1/18 12: 1/36 Squares 1: 1 2: 4 3: 9 4: 16 5: 25 6: 36 7: 49 8: 64 9: 81 10: 100 11: 121 12: 144 13: 169 14: 196 15: 225 16: 256 17: 289 18: 324 19: 361 20: 400 Bayes’ theorem P(A|B) = P(B|A) P(A) / P(B)