Unformatted text preview: COMM 290 Introduc1on to Quan1ta1ve Decision Making PROBABILITY Introduc1on to Probability Quan/ta/ve Decision Making Linear Program • Formula1on • Op1miza1on • Sensi1vity analysis Probability • Basic rules • Joint probability, condi1onal probability, Bayes Theorem Decision Analysis • Decision tree • Value of informa1on Random Variables • Expected value and variance • Independent and dependent random variables 1 This class • Making reasonable assump1ons – Train problem – Picking a ball – LoQery • Simula1on with Excel – Island problem Train Problem Each day, a man at a train sta1on can either take a westbound train and visit his sister in Town A or he can take an eastbound train and visit his brother in Town B. Westbound and eastbound trains leave every 20 minutes and each day the man arrives at random 1mes at the train sta1on. He takes the ﬁrst train that comes along and visits either his sister or brother. AWer many months of doing this on a daily basis, what percentage of the 1me do you think he visits his brother? his sister? Comfortable with your answer? Suppose aWer many months of travelling on these trains, the man is aware that he has visited his brother about 75% of the 1me compared to visi1ng his sister only 25% of the 1me. How is this possible? 2 Train Problem: Solu1on • Rare event? • Train schedule? Eastbound 7, 7:20, 7:40 etc. Westbound 7:05, 7:25, etc. E W 7:00 E W 7:10 7:20 E W 7:30 7:40 7:50 Making reasonable assump/ons is important Pick One Ball from the Bowl What is the probability it will be white? • With reasonable assump1on, 70% [theore1cal answer] 3 Simula1on with Excel • =RAND() W R 0 1 • =IF(logical_test, value_if_true, value_if_false) – E.g., =IF(G4<0.7, "W","R") • Condi1onal formaing Island Problem The government of a small island community is considering a policy that will control the popula1on on the island. An experimental policy that they are considering is to require couples to have children, but they must stop as soon as they have a girl. When all 50 families have stopped having children, we want to answer some ques1ons. 4 Island Problem (Con1nued) Ques/ons 1. Would you expect there to be more boys on the island than girls? more girls than boys? equal numbers of boys and girls? Why? 2. What would you expect the popula1on of the island to be? 3. What will be the maximum # of girls in a family? 4. What will be the maximum # of boys in a family? 5. What will be the average number of girls in each family? 6. What will be the average number of boys in each family? 7. What will be the average number of children in each family? 8. What do you think is the objec1ve of this policy? Do you think the objec1ve has been met? Island Problem (Con1nued) Assump/ons 1. The probability of having a boy is equal to the probability of 2.
4. 5. having a girl. Probabili1es are independent; that is, for any par1cular family, the probabili1es do not change between successive children. There are no deaths on the island. There are only two genera1ons under considera1on. In other words the 1me frame is rela1vely short and therefore children born under this policy do not have children of their own. Since the government is trying to control the popula1on, the assump1on is that all families want to have as many children as possible. Therefor it is assumed that a family could not, for example, have one boy and then stop. 5 Summary • Making reasonable assump1ons is important • With reasonable assump1ons, we can calculate simple probabili1es • Excel can be used to simulate experiment 6 ...
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- Fall '19
- – Island