challenge Problem proble put your u erstanding of the concepts learned in Detail

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challenge Problem proble put your u erstanding of the concepts learned in Detailed hints and tips are provided to help you along, but this problem e a good amount of time and effort on your part. Be patient and don't be di ouraged if you don't et it right awa problem designed to challenge you are you up for the challenge Scenario 1: Kathy drives down Hillsborough Street on Tuesday and Wednesday. Suppose that on any given day there is a 44% chance that she will have to stop for the stoplight at Brooks Av Hints and Tips Draw a tree diagram for this scena io. [Hint: Do this by branching up first to ops on Tuesday" and "doesn't stop on Tuesday en from each of those branch up to "sto ps on Wednesday and "doesn't stop on Wednesday. Figure out which numbers need to be inserted on each of the branches and m ply up" to find the probabilities for each of the fol Note: In order to be uccessful with Scena os 2 and 3 it mportant to fully unders nd ho the Scenari o 1 sing the tree diagram described Use this tree to answer the following questio s. (Give your answers correct to three decimal places. (a) What is the probability that she will have to stop at the stoplight on Tuesday? What is the probability that she have to stop at the stoplight on both Tuesday and Wednesday c) What is the probability she has to stop fo e stop light on exactly one of the two days 493 Scenario 2: April drives down Hillsborough Street on Thursday and Friday. Suppose you know that the probability she has to stop for the stoplight at Brooks Ave, on both days 0.44. What is the probability that April will have to stop on any given day? Hints and Tips ee diagram. The overall layout of the tree wil look similar to your tree for information to place any numbers in your tree. To overcome this, we will assign a variable ("p") to Again, you will need to draw a Scenario however, this time you are not given enough be equivalent to P(stops on any given day Place "p" in your tree in the appropriate place Since we have P(stops on any given p", this means that PCdoe stop on a given day) must be e all to p Place p in your tree in e appropriate lay places. Using algebra, "multiply up" your branches to find the probability of each of the four outcomes (if you do this correctly each probability will be a mathematical statement with "p" in it Now that you have your tree diagram h al he probabilities in te rms of "p", look at he information that you were give "suppose you know that the probabiWty she has to stop for the stoplight at Brooks Ave. on both days 0.44 Where would this 0.44 go in your tree? Use that info rmation to write an algebraic eq ation. Solve this equation for "p' What is the probability that April wi have to stop on any given day? Or, equivalently, wha the exa value of p in this Give your answer correct to three decimal places. Scenario 3 Becky drives down Hillsborough Street on Saturday and Sunday. Suppose you know that the probability she has to stop for the stoplight at Brooks Ave. on exactly one of these days is 0.32. What is the probability that Becky will have to stop o any given day? Hints and Tips st as in Scenario 2, you are not given P(stops on any given day). Therefore, you'll need to draw a tree utilizing "p" and "1. p' Once you have your tree dia am with all the probabilities in terms of p ook at the information that you were given. "Suppose you know that the probabiity she has to stop for the stoplight at Brooks Ave. on exactly one of these days is 0.32. Notice ere is n one outcome in ctly one day. Yo need to add the probabilities of branches together if yo o fil d the overall P(Becky stops on e y one day). Use that informatio n and you Your tree that opping on e 0.32 value to writ e an algebraic equation. Solve this equation fo Recall from previous math courses that o solve an equation of this nature you need to use the quad ic formula. If you've forgotten the formula simply do a Google search fo quadrat ion solver here are lots of good, free ones out there for you to use. You have to get e equation in e right format, but once you do that you n use a solver to do all calculations. ou'll find that there are act y two correct answers to this question This means that there are two pot ble values of "p" that could result in P(Becky stops at Brooks Ave, on exactly one of e two days 0,32 have to stop on any given day? Or, equivalently, what are the exact values of "p" in this scenario What is the probability at April to three decimal places. ve your an smaller value larger value

Explanation / Answer

Scenario1.

Probability of stopping on tuesday = 0.44

   Probability of not stopping on tuesday = 0.56

   Probability of stopping on wednesday = 0.44  

Probability of not stopping on wednesday = 0.56

b) Probability that she has to stop both Tuesday and wednesday= 0.44 X 0.44 = 0.1936 = 0.194

Scenario4:

Probability of A winning a game = 0.64. Therefore, probability of A losing the game = 1-0.64 = 0.36

Probability of B winning a game = 0.36. Therefore, probability of B losing the game = 1-0.36 = 0.64

There are 7 independent games. Hence the number of trials = 7

Probability of one of the team wins 3 games and but the other team wins the series = Probability that A wins 3 games but B wins the series +  Probability that B wins 3 games but A wins the series

The situation follows binomial distribution with n= 7 p, probability of success = .64 for A and 0.36 for B

q = probability of failure = .36 for A and 0.64 for B

Therefore the required probability = 7C3 (.64)3 (.36)4 + 7C3 (.36)4(64)4

= 0.0044 +0.2783

= 0.2783

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