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Q: Probability examples - for practice ( Answered ,   0 Comments )
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 Subject: Probability examples - for practice Category: Science > Math Asked by: curiousmaz-ga List Price: \$5.00 Posted: 30 Sep 2005 23:26 PDT Expires: 30 Oct 2005 22:26 PST Question ID: 574902
 ```Can anyone help with some practice questions? Much appreciated - haven't done math in years, just getting back into it. Q1. A company is about to launch a new beverage. In the past 40% of its beverages have successfully launched. Before any soft drink is launched, the company conducts research and receives a report predicting favourable or unfavourable sales. In the past, 80% of successful bevs and 30% of unsuccessful bevs received favourable reports. What is the probability this new bev will be successful IF it receives a favourable report? Q2. A vendor at a baseball game needs to choose whether to sell soft drinks or hotdogs on the day of the game, depending on weather. The dollar profits for different products under different conditions are set out below and reflect pricing strategies - COLD DAY, HOT DOG PROFITS = \$50 COLD DAY, SOFT DRINK PROFITS = \$30 HOT DAY, HOT DOG PROFITS = \$60 HOT DAY, SOFT DRINK PROFITS = \$90 Based on past experience at the same time of year, the vendor estimates probability of it being a hot day at 0.40. Prior to deciding he listens to the weather. The forecaster says it will be cold. In the past, when it has been cold, the weather has forecasted cold 70% of the time. When it has been hot, the forecaster has been correct 60% of the time. Hence: - what is the probability the weather will be cold today given the forecast of cold? - what is the probability the weather will be hot given the forecast of hot? - Given the forecast is for cold weather - what should the vendor sell? [Does this require a decision tree???] Q3. A retailer buys mushrooms for \$2/kg and sells them for \$5/kg. Mushrooms not sold on the first day are returned to the wholesaler who buys them back at: \$2 for 1 kg, \$1.75 for 2kg, \$1.50 for 3 kg, \$1.25 for 4 or more kg. A 90 day observation of past demand shows: (imagine this is a table) DAILY DEMAND in KG = 10 11 12 13 NUMBER OF DAYS = 9 18 36 27 - Calculate the expected daily demand for the mushrooms? - Based on comparison of expected profits, would you recommend the retailer orders 12kg or 13kg of mushrooms per day? You can assume the customer requests for mushrooms that can't be met on the day do not affect future demand.```
 ```For these problems, you will need a basic understanding of Bayes' Theorem (conditional probability) and of the concept of expected value. Decision trees use the idea of expected value to graphically represent each condition and choice. I hope these are covered in your textbook? I will briefly go over them. Bayes' theorm states that he probability of A given B equals the probability of B given A times the probability of A, divided by the probability of B. Also written P(A|B) = P(B|A) * P(A) / P(B) Expected value is the sum over all possible outcomes, of the value of an outcome multiplied by the probability of that outcome. For example, imagine a 100-ticket raffle with a payout of \$50. The ?expected payout? is 0.01*\$50 + 0.99*\$0 = \$0.50. So, if an organization sells 100 tickets at \$1 each, they will turn a profit. For a tutorial, I recommend the following links: http://sysopmind.com/bayes/bayes.html A humorous and readable introduction to Bayes Theorem. This includes Javascript calculators in the text, so that you can follow along with the examples. http://members.aol.com/johnp71/bayes.html A Bayes theorem calculator. http://www.mindtools.com/pages/article/newTED_04.htm A nice approach to decision trees that includes calculation of expected values, and probabilities of an outcome. Q1. Using Bayes' Theorem, the probability of this beverage being successful, given that it received a favorable report, is the probability of receiving a successful report given that the beverage was successful, times the probability of being successful, divided by the probability of receiving a favorable report. P = 0.8 * 0.4 / (0.4*0.8 + 0.6*0.3) = 0.64 Another way to think about it, is to calculate all four possibilities first: Successful + favorable report = 0.4 * 0.8 = 0.32 Unsuccessful + favorable report = 0.6 * 0.3 = 0.18 Successful + unfavorable report = 0.4 * (1 ? 0.8) = 0.08 Unsuccessful + unfavorable report = 0.6 * (1 ? 0.3) = 0.42 They sum to 1, so my math is correct! The fraction of successful beverages among those receiving a favorable report is 0.32/(0.32+0.18) = 0.64 Q2. Predicted Cold + Cold = 0.7 * 0.6 = 0.42 Predicted Cold + Hot = 0.7 * (1 ? 0.6) = 0.28 Predicted Hot + Hot = 0.6 * (1 ? 0.6) = 0.24 Predicted Hot + Cold = 0.6 * 0.6 = 0.36 The probability that it will be cold given that it is forecast to be cold is the probability that it was forecast to be cold given that it is cold, times the probability that it will be cold, divided by the probability that it will be forecast to be cold -- 0.42 / (0.42 + 0.28) = 0.6 - what is the probability the weather will be hot given the forecast of hot? The probability that it will be hot given that it is forecast to be hot is the probability that it was forecast to be hot given that it is hot, times the probability that it will be hot, divided by the probability that it will be forecast to be hot. Deciding which product is more profitable to sell requires calculating the expected value of both choices. A decision tree works for this, although a table in a spreadsheet works also. The expected value is the profit from a particular product and situation, multiplied by the probability that that situation will actually occur. Given the prediction for cold weather, the probability of cold is 0.6. Likewise, the probability of hot weather is 0.4. The expected value of selling hotdogs is 0.4*\$60 + 0.6*\$50 = \$54 The expected value of selling softdrinks is 0.4*\$90 + 0.6*\$30 = \$54 The vendor should use factors other than the weather to decide whether to sell hotdogs or softdrinks. Perhaps it is easier to carry hotdogs? Perhaps softdrinks help him stick to his diet? Perhaps he prefers a risk ? if so he should sell softdrinks, since in reality he will either make a large profit or a small one. The expected profit, based on the weather, is the same either way. Q3. Again, the expected demand is the sum, over all levels of demand, of the product of a demand of N kg multiplied by the probability of that level of demand. The probability of a particular level of demand is the number of days at that demand level, divided by the total number of days. Thus expand your table: P-DEMAND 0.1 0.2 0.4 0.3 The expected demand level is 0.1*10 + 0.2*11 + 0.4*12 + 0.3*13 = 11.9 kg of mushrooms. A decision tree could help the retailer decide how many mushrooms to buy; a table showing the expected profit from each combination, and a calculation of the profit from each purchase volume is also effective. Post your answer in a request for clarification, and I will let you know my answer. I used the Wikipedia's definition of Bayes' Theorem for reference. http://en.wikipedia.org/wiki/Bayes'_theorem I hope you find this helpful, and can solve problems after seeing some examples. If you have any questions, feel free to ask for a Request for Clarification. And, let me know how many mushrooms you think the retailer should buy, and I will let you know what I think. ;)``` Request for Answer Clarification by curiousmaz-ga on 02 Oct 2005 02:40 PDT ```Thanks for this neurogeek. I am very rusty on this stuff, but I am working on the decision tree and mushroom problem. One thing I did seem to miss was how you arrived at: "P-DEMAND 0.1 0.2 0.4 0.3". I'llcheck in with my suggested solution after, I have a few others to do in the meanwhile. BTW - the sysomind link on Bayesian thinking is a very good explanation, and entertaining. Thanks again.``` Request for Answer Clarification by curiousmaz-ga on 02 Oct 2005 02:45 PDT ```Oh sorry, I re-read your response. Of course you got the P(demand) figires from the table, you just mixed up 0.4 with 0.3. No big deal.I'll check back soon.``` Request for Answer Clarification by curiousmaz-ga on 02 Oct 2005 03:05 PDT ```Mushrooms/kg Buy Price Sell Price Profit per day 10 KG \$20 \$50 \$30 11 KG \$22 \$55 \$33 12 KG \$24 \$60 \$36* 13 KG \$26 \$65 \$39 BTW - the figures were askew in your answer. I re-did them: (0.1*10) + (0.2*11) + (0.3*12) + (0.4*13) = 12 kg So - with expected demand at 12kg per day, then the mushroom retailer could safely order 12kg and avoid a loss on any stock ordered which has to be returned or sold to the wholesaler at a reduced rate. If they wanted to take a risk to earn \$39 in profit, they could order 13kg - if one kg has to be returned unsold (vbecause exp. demand is 12kg) then they'd minus \$2 off their \$39 profit. At \$37 they're ahead with 13kg. I thought the decision tree was helpful but not so necessary for this problem. The table was more approp. in this case. let me know what think.``` Clarification of Answer by neurogeek-ga on 02 Oct 2005 08:34 PDT ```Hello again curiousmaz, I'm glad you liked that Bayes Theorem explanation. I think it has enough substance to make up for its saccharine enthusiasm. It looks like you are on the right track now, in understanding this type of problem. I thought that the demand for 12 kg occured on 36 days, and the demand for 13 occured on 27, which is how I got the probabilities reversed. Also, I think he recieves a REFUND for the unsold mushrooms -- so, if he buys 13 kg, but only sells 12 his records will look like this: 10/1/2005 purchased 13 kg mushrooms -\$26 10/1/2005 sold 12 kg mushrooms +\$60 10/2/2005 returned 1 kg mushrooms +\$2 Net profit (2+60-26) = \$36 I made a table for both cases, showing the demand paired with the profit: BOUGHT 12 kg DEMAND 10, profit 29.5 DEMAND 11, profit 33 DEMAND 12, profit 36 DEMAND 13, profit 36 Expected Profit = 0.1*29.5 + 0.2*33 + 0.3*36 + 0.4*36 = 34.75 BOUGHT 13 kg DEMAND 10, profit 28.5 DEMAND 11, profit 32.5 DEMAND 12, profit 36 DEMAND 13, profit 39 Expected Profit = 0.1*28.5 + 0.2*32.5 + 0.3*36 + 0.4*39 = 35.75 Let me know if anything else is unclear. --neurogeek``` Request for Answer Clarification by curiousmaz-ga on 02 Oct 2005 21:33 PDT `Thanks for your efforts! The E.D is also helpful. All clear, Ciao.` Clarification of Answer by neurogeek-ga on 04 Oct 2005 16:22 PDT `Thanks for the tip!`
 curiousmaz-ga rated this answer: and gave an additional tip of: \$2.00 ```A helpful method of answering which enabled appropriate explanation and problem solving but didn't give the final answers away! Thanks. Wish I had more money to tip with - this is just a small token.```