The number of goals that J scores in soccer games that her team wins is Poisson distributed with mean 2, while the number she scores in games that her team loses is Poisson distributed with mean 1. Assume that, independent of earlier results, J’s team wins each new game it plays with probability a. Find the expected number of goals that J scores in her team’s next game. b. Find the probability that J scores 6 goals in her next 4 games. Hint: Would it be useful to know how many of those games were won by J’s team. Suppose J’s team has just entered a tournament in which it will continue to play games until it loses. Let X denote the total number of goals scored by J in the tournament. Also, let N be the number of games her team plays in the tournament. a. Find E[X]. b. Find P ( X = 0 ) . c. Find P ( N = 3 | X = 5 ) .
The number of goals that J scores in soccer games that her team wins is Poisson distributed with mean 2, while the number she scores in games that her team loses is Poisson distributed with mean 1. Assume that, independent of earlier results, J’s team wins each new game it plays with probability a. Find the expected number of goals that J scores in her team’s next game. b. Find the probability that J scores 6 goals in her next 4 games. Hint: Would it be useful to know how many of those games were won by J’s team. Suppose J’s team has just entered a tournament in which it will continue to play games until it loses. Let X denote the total number of goals scored by J in the tournament. Also, let N be the number of games her team plays in the tournament. a. Find E[X]. b. Find P ( X = 0 ) . c. Find P ( N = 3 | X = 5 ) .
Solution Summary: The author calculates the probability that J scores 6 goals in her next 4 games.
The number of goals that J scores in soccer games that her team wins is Poisson distributed with mean 2, while the number she scores in games that her team loses is Poisson distributed with mean 1. Assume that, independent of earlier results, J’s team wins each new game it plays with probability
a. Find the expected number of goals that J scores in her team’s next game.
b. Find the probability that J scores 6 goals in her next 4 games.
Hint: Would it be useful to know how many of those games were won by J’s team. Suppose J’s team has just entered a tournament in which it will continue to play games until it loses. Let X denote the total number of goals scored by J in the tournament. Also, let N be the number of games her team plays in the tournament.
Starting with the finished version of Example 6.2, attached, change the decision criterion to "maximize expected utility," using an exponential utility function with risk tolerance $5,000,000. Display certainty equivalents on the tree.
a. Keep doubling the risk tolerance until the company's best strategy is the same as with the EMV criterion—continue with development and then market if successful.
The risk tolerance must reach $ 160,000,000 before the risk averse company acts the same as the EMV-maximizing company.
b. With a risk tolerance of $320,000,000, the company views the optimal strategy as equivalent to receiving a sure $____________ , even though the EMV from the original strategy (with no risk tolerance) is $ 59,200.
Starting with the finished version of Example 6.2, attached, change the decision criterion to "maximize expected utility," using an exponential utility function with risk tolerance $5,000,000. Display certainty equivalents on the tree.
a. Keep doubling the risk tolerance until the company's best strategy is the same as with the EMV criterion—continue with development and then market if successful.
The risk tolerance must reach $ ____________ before the risk averse company acts the same as the EMV-maximizing company.
b. With a risk tolerance of $320,000,000, the company views the optimal strategy as equivalent to receiving a sure $____________ , even though the EMV from the original strategy (with no risk tolerance) is $ ___________ .
A television network earns an average of $14 million each season from a hit program and loses an average of $8 million each season on a program that turns out to be a flop. Of all programs picked up by this network in recent years, 25% turn out to be hits and 75% turn out to be flops. At a cost of C dollars, a market research firm will analyze a pilot episode of a prospective program and issue a report predicting whether the given program will end up being a hit. If the program is actually going to be a hit, there is a 75% chance that the market researchers will predict the program to be a hit. If the program is actually going to be a flop, there is only a 30% chance that the market researchers will predict the program to be a hit.
What is the maximum value of C that the network should be willing to pay the market research firm? Enter your answer in dollars, not in million dollars.
$ __________
Calculate EVPI for this decision problem. Enter your answer in dollars, not in million…
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.
Discrete Distributions: Binomial, Poisson and Hypergeometric | Statistics for Data Science; Author: Dr. Bharatendra Rai;https://www.youtube.com/watch?v=lHhyy4JMigg;License: Standard Youtube License