Consider a betting system with two possible outcomes labelled 1 and 2, and two possible wagers, also labelled 1 and 2. The returnfunction is given by r₁ (1) = 2.4, r₁ (2) = 3.3, r₁ (2) = x, and r₂(2) = 0.7, where x is a real number. You are told that the bettingsystem is fair, that is, there is no arbitrage. Determine x correct to 3 significant digits.Answer:

Step 1: Set Up Fairness Equations Step 2: Solve for x Using p1

Answered: Consider a betting system with two…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Three balanced coins are tossed independently. One of the variables of interest is Y₁, the number of heads. Let Y₂ denote the amount of money won on a side bet in the following manner. If the first head occurs on the first toss, you win $1. If the first head occurs on toss 2 or on toss 3 you win $2 or $3, respectively. If no heads appear, you lose $1 (that is, win -$1). a Find the joint probability function for Y₁ and Y2. b What is the probability that fewer than three heads will occur and you will win $1 or less? [That is, find F(2, 1).]
Given P(A) = 0.21, P(B) = 0.24, and P(B|A) = 0.24, are A and B independent or dependent?
The following dice game is offered to you. You may simultaneously roll one red die and three blue dice. The stake is $1. If none of the blue dice matches the red die, you lose your stake; otherwise, you get anyway paid k +1 dollars if exactlyk of the blue dice match the red die. In the case that exactly one blue die matches the red die, you get paid an additional $0.50 if the other two blue dice match. the median payoff of the game is
Suppose you are deciding whether or not to bring an umbrella to school and the weather can be either sunny or rainy. Initially, you think the probability that it will be sunny is P(S)=0.6, and the probability that it will be rainy is P(R)=0.4. If you bring an umbrella and it rains, your payoff is 6. If you bring an umbrella and it is sunny, your payoff is 9. If you don’t bring an umbrella and it rains, your payoff is 0. If you don’t bring an umbrella and it is sunny, your payoff is 10. Suppose that you check two websites and see that both say it will be sunny. You know that the two forecasts are independent of each other, and each forecast is correct 70% of the time. That is, the probability that the website says it will be sunny given that it actually will be sunny is 0.7, and the probability that the website says it will rain given that it actually will rain is 0.7. If you are a Bayesian updater, what is your updated belief about the probability that it will be sunny? Do you bring…
A casino offers the following game, which costs 10 dollars to play. Two cards are selected at random without replacement from a standard deck of 52 cards. If both of the cards are kings, you win 220 dollars and get your 10 dollars back. If exactly one of the cards is a king, you win 20 dollars and get your 10 dollars back, Otherwise, you lose the 10 dollars. Let X be equal to the amount of money won as the result of playing this game one time, where winning a negative amount is equivalent to losing that amount. Find P(X>0) Find E(X)
Avicenna, an insurance company, offers five-year commercial property insurance policies to small businesses. If the holder of one of these policies experiences property damage in the next five years, the company must pay out $23,600 to the policy holder. Executives at Avicenna are considering offering these policies for $791 each. Suppose that for each holder of a policy there is a 3% chance they will experience property damage in the next five years and a 97% chance they will not. (If necessary, consult a list of formulas.) If the executives at Avicenna know that they will sell many of these policies, should they expect to make or lose money from offering them? How much? To answer, take into account the price of the policy and the expected value of the amount paid out to the holder. O Avicenna can expect to make money from offering these policies. In the long run, they should expect to make dollars on each policy sold. O Avicenna can expect to lose money from offering these policies.…
Avicenna, an insurance company, offers five-year commercial property insurance policies to small businesses. If the holder of one of these policies experiences property damage in the next five years, the company must pay out $26,500 to the policy holder. Executives at Avicenna are considering offering these policies for $497 each. Suppose that for each holder of a policy there is a 2% chance they will experience property damage in the next five years and a 98% chance they will not.(If necessary, consult a list of formulas.) If the executives at Avicenna know that they will sell many of these policies, should they expect to make or lose money from offering them? How much? To answer, take into account the price of the policy and the expected value of the amount paid out to the holder. Avicenna can expect to make money from offering these policies. In the long run, they should expect to makedollars on each policy sold. Avicenna can…
a) When you bet on a race-horse with odds of m-n, you stand to win m rands for every bet of n rands if your horse wins; for instance, if the horse you bet is running at 5-2 and wins, you will win R5 for every R2 you bet (i.e. R2 bet will return R7). You are first required to compensate an amount of R5 as an entrance fee in order to bet on a race-horse. Here are some actual odds from a 1992 race at Belmont Park. The favourite at 6-2 was Pleasant Tap. The second choice was Thunder Rumble at 7-2, while the third choice was Strike the Gold at 3-1. Assume you are making a R10 bet on one of these horses. The payoffs are your profits. (If your horse does not win, you lose your entire bet and the entrance fee. Of course, it is possible for none of your horses to win). Set up the reward payoff matrix.
Suppose you are deciding whether or not to bring an umbrella to school and the weathercan be either sunny or rainy. Initially, you think the probability that it will be sunny isP (S) = 0.6, and the probability that it will be rainy is P (R) = 0.4. If you bring an umbrellaand it rains, your payoff is 6. If you bring an umbrella and it is sunny, your payoff is 9. Ifyou don’t bring an umbrella and it rains, your payoff is 0. If you don’t bring an umbrellaand it is sunny, your payoff is 10. Suppose that you check two websites for weather forecasts.You know that the two forecasts are independent of each other, and each forecast is correct70% of the time. That is, the probability that the website says it will be sunny given that itactually will be sunny is 0.7, and the probability that the website says it will rain given thatit actually will rain is 0.7. Suppose that you suffer from confirmation bias. You initially thinkit is more likely to be sunny, so you misinterpret rain forecasts as sun…
Rhett owns a cupcake bakery and is analyzing his sales of cupcake delivery orders. Based on his daily sales of delivery orders for the past month, he has already calculated the probabilities, ?(?),P(X), for the number of boxes of cupcakes, ?,X, purchased by a single customer in a single day, as shown in the table. ?X 1 2 3 4 5 ?(?)P(X) 0.75 0.05 0.10 0.05 0.05 Calculate the mean number of boxes of cupcakes, ??,μX, sold and delivered per person in a single day. Express your answer to two decimal places. ??=μX= If a box of cupcakes costs $40.00$40.00 and the flat-rate delivery fee is $7.00$7.00, calculate the mean sales per person, ??,μY, that the cupcake shop makes in a single day for delivery orders. Express your answer to the nearest cent. ??=$μY=$
Suppose you are deciding whether or not to bring an umbrella to school and the weathercan be either sunny or rainy. Initially, you think the probability that it will be sunny isP(S)=0.6, and the probability that it will be rainy is P(R)=0.4. If you bring an umbrella and itrains, your payoff is 6. If you bring an umbrella and it is sunny, your payoff is 9. If you don'tbring an umbrella and it rains, your payoff is O. If you don't bring an umbrella and it is sunny,your payoff is 10. Now suppose that you suffer from confirmation bias. You are pretty sure that it is going tobe sunny, so you misinterpret rain forecasts as sun forecasts 20% of the time but alwayscorrectly interpret sun forecasts. You are aware of your own confirmation bias. If yourinterpretation is that you have seen two sun forecasts, what is your updated belief about theprobability of sun? Will you bring your umbrella?
You are playing a game of dice where it involves rolling three dice and betting on one of the six numbers that are on the dice. The game costs P80 to play. You win P160 if the number you bet appears on only one dice, P180 if on two dice and P320 if on all three. What is your expected profit from the game?

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