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A. Show how you would use the division, multiplication, and addition probability rules to calculate the probability of tossing a coin 4 times and getting only 1 head. Note you may not need to use all the rules! Show your work. (2 pts)

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Karol E.
Karol E.Lv8
2 Apr 2021
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Related textbook solutions

Introductory Statistics
OER Edition, 2013
Openstax
ISBN: 9781938168208

Related questions

Alice and Bob vote in each parliamentary election. If, in a certain election, Alice and Bob vote for the same party, they vote for it again in the next election. If they vote for different parties, next each of them switches their opinion independently with probability 1/4 .

 

1. Devise a Markov model to describe their votes in the n-th election, under the simplifying assumptions that they are immortal, the parliamentary system in the UK is stable and there are only two parties (C and L). 

 

2. Draw the transition graph and write down the transition matrix (note that you need to number the states!) 

 

3. Assume that in the n-th election Alice voted C and Bob voted L. What is the (conditional) probability that in the n + 2-nd election Alice will vote L and Bob will vote C? 

 

4. Assume that in the n-th election Alice and Bob both voted L. What is the (conditional) probability that in the n − 1-st election Alice voted C?

Happy Clown Carnival Company is scheduled to appear in a city on a particular Saturday. The profits that will be obtained depend heavily upon the weather. If the weather is rainy, the carnival loses $15,000; if it is not rainy, the carnival makes a profit of $5,000. Historically, the probability of rain on any given day is 10%.

The carnival has to set up equipment for its show, but it can cancel the show prior to setting up its equipment. Cancelling the show results in a loss of $1,000, regardless of the weather. However, for an additional $250 cost, the carnival can postpone its setup decision until the Friday before the scheduled performance. At that time, the carnival can obtain the local weather forecast for Saturday’s weather.

The Weather Bureau has compiled data on its past forecasts. When it was rainy, the Weather Bureau had predicted rain 74% of the time. When it was not rainy, the Weather Bureau had predicted no rain 86% of the time.

(a) Set up the decision tree and determine the decision (SETUP or CANCEL) that maximizes the expected profit without waiting for a weather forecast on the blank page provided. Leave enough space to add to this tree in part (d).(15 pts for correct tree, 12 pts for decision and backup calculations.)

(b) Determine the Expected Value of Perfect Information (EVPI) for the situation depicted in your decision tree from (a).(7 pts.)

(c) Append to your decision tree in (a) a decision tree diagram that includes waiting for the results of the weather forecast. Be sure to clearly label the branches and nodes. (15 pts.)

(d) Determine the two posterior probability distributions for Saturday’s actual weather given the weather forecasts on Friday and put your results in the appropriate places on your tree in (c). (Hint: Bayes. ). (15 pts.)

(e) . Calculate the Expected Values at each appropriate node for the part of the tree diagram you added in (c). (30 pts.)

(f) Based on your results in (e), what is your optimal decision strategy? (5 pts.)

(g) Calculate the Expected Value of Sample Information (EVSI) and the efficiency of sample information for this situation. Also calculate the Expected Net Gain of Sample Information. (ENGSI is the amount you expect to gain from the sample information after accounting for what it cost you to get that information.) (5 pts.) Use this page for your decision tree. Please write clearly enough so I can tell what your answers are.

Rules of dice game:

• All players start with 10 points

• A dice is rolled, and if the number on the dice is 1, 2, 3, 4 or 5, this number is added to each player's score. If the number is 6, the score off all players is reduced to zero

• Before the dice is rolled each player is given the choice of continuing or taking their current score as their final score.

• This process is repeated until a 6 is rolled and the game ends. All players still playing at this point receive zero as their final score.

Suppose that you choose to play the first dice roll.

(i) Compute the expected value and standard deviation of your score after the first dice roll.

(ii) On average, do you do worse or better than someone who choose not to play and take 10 as their final score?

Now suppose that a 6 was not rolled and you chooses to continue playing

(iii) On average, what score would you have before the dice is rolled?

(iv) Compute the expected value for your score after the second dice roll.

(v) On average, do you do worse or better than someone who choose to stop playing after 1 dice roll?

(vi) Is there an optimal number of dice rolls you should continue playing for? Explain your reasoning.

(vii) If the rules are changed such that when the number is 2, 3, 4, 5 or 6, this number is added to your score. While if the number is 1, your score is reduced to zero. Discuss how this affects the optimal number of dice rolls you should continue playing for.

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