Friday, July 12, 2013

Math in Jeopardy!!!

This morning, I went down to the lobby at 7 AM, and went with the three-mile run group. This time we did not stop. I survived, but ended up being at the back end of group. I then hurried to take a shower and got to the dining commons by 8. Today was supposed to be our Proctor Group Breakfast. (As you may have guessed, all of us eating breakfast at the same table, and talking.) This plan was a major backfire. First of all, our proctor was not even there at 8. Secondly, only about half our group (six of us) were on time. By 8:20, all but one person had showed up, but many had already left. The idea was good, but next time we do this, we'll be sure to do lunch or dinner, as everyone has to be in the commons, and is more awake/talkative.

Today's Puzzler was about probability. Three guys are holding paintball guns, and are standing in a triangle. Their names are Andy, Bob, and Charley. Andy has 30% accuracy, Bob never misses, and Charley has 50% accuracy. They shoot in order (Andy, Bob, Charley). If you are hit, you are out and can no longer shoot. Assuming everyone opts for their optimal strategy, what is Andy's best move? As always, STOP reading if you want to solve it yourself.

Andy has three options for his first turn. He can hit Bob, hit Charley, or miss. If Andy hits Bob, Charley will shoot at Andy. There is a 50% chance Charley will miss. If Charley misses, then Andy gets another shot. Andy has a 30% chance of hitting Charley and missing. This can continue of indefinitely, although that is unlikely. Since you are multiplying by a number with an absolute value of less than one, you can use the formula S = 1st term/(1-change factor). The first term is 0.5*0.3, and the change factor is 0.5*0.7, so Andy has a 23.07% chance of winning with that route. If Andy hits Charley, its Bob's turn, and Andy is the only one left, so Bob shoots Andy, making Bob the victor. This means Andy has a 0% chance of winning if he hits Charley. If Andy misses, then Bob will hit Charley, and then its Andy's turn. Andy then has a 30% chance of hitting Bob and winning. If Andy misses, Bob will hit Andy and win. Therefore, Andy's best option is to intentionally miss his first shot to get a chance at shooting Bob later. If you calculate the probability of each person winning, you find the following: Andy 30%, Bob 70%, and Charley 0%. Then we figured out what would have to change to give Charley a chance at winning. This turned out to be his accuracy percentage. The only way for Charley to win is to get Andy to hit Bob in his first shot. The only way to do that is to make it more likely that Andy will win if he shoots Bob, then if he misses intentionally. We calculated that Charley's accuracy would have to go down to 40% to make this happen. Since we spent almost an hour and a half solving and discussing this problem, we did not do independent study today.

Today's lecture was titled "Probability Curiosities." Dawson started with a story. There once was a Chemistry Professor from Duke, who had a very challenging class. Two of his students were doing really well, and there was a final on Monday morning. The two students decided they did not need to study, so they drove to a party in another state. They stayed Saturday night, and decided to stay Sunday night too, because they did not feel like taking the final. On Monday afternoon, they ran to the professor's office, and asked him for forgiveness, saying they got a flat tire. The professor agreed to let them take the test the next day at 8. They sit down in the room, and the teacher puts them in opposite corners. Then they start the final. The first question is a regular Chemistry question worth 10 points. They answer it, and more on to question two which is worth 90 points. It reads, "Which tire was it?" Dawson had us calculate the probability that the students would pass the final. It turned out to by 25%, because the teacher does not know the right answer, he only knows if they are lying if the answers do not match. The first person has 4 possible answers, and the second person has only one possible answer. Therefore, their are four favorable outcomes out of the 16 possibilities, or 25%.

Next Dawson had us play a probability game. We were each given a quarter. We had to flip the quarter until we got tails. We were supposed to do this 10 times, so eventually we would have gotten 10 tails and an unknown number of heads. He also put a dollar amount on each. Getting just a T gets you $2. Heads and then tails is $4, HHT is $8, HHT is $16, etc. He then had us calculate the amount of money we would earn from the game based on our flips. He then put all our results in a spread sheet and found the average. We averaged a $20 payout per game. This is why a game like this does not exist at carnivals. No one wants to risk over $20, when there is a 50-50 chance of getting $2 back.

Next we calculated the odds of winning in the game show "Let's Make a Deal." In this game, there are three doors. Behind one of the doors, there is a new car. Behind the other two are goats. The contestant picks a door. The host then opens a door with a goat behind it (not your door.) He then asks if you want to change your door to the other hidden one. We got in pairs and played this game. One person was Stick (they stuck with their original choice) and the other was Switch (they always switched their door when asked). Dawson taught us how to use or calculator to generate a random whole number between 1 and 3. The "host" (the person not playing) used this to pick which door the car was behind. Each person played 10 games. It turns out that you have better chances of winning if you switch. We compiled all our data, and found that Stick won 33% of the time, and Switch won 66% of the time. We then went over the logic behind it. If you pick a door and stick with it, there is a 33% chance it is a car, and a 66% chance it is a goat. Since the host always reveals a goat, that narrows it down to two doors that could have a car. This doubles your chances of winning, from 33% to 66%. I had never heard of this game show before, but I understand why people stick with their door. They are nervous and scared, so they don't think about the statistics behind it. I think learning things like this is really beneficial to our lives. Yes, we are most likely not going to be on a game show, but it teaches us to look at the probability behind everything, despite what our gut tells us.

Next Dawson gave us each an index card, and asked us to calculate what we think is the probability that two people in our class share the same birthday. On the other side of the card, he had us calculate how many people would be needed to get a 50% chance of a match. He did not teach us how to do this, so this was really just a guess. Next, he showed us how to calculate this. Since this can be very complicated, it is easier to find the probability that no one shares a birthday, and subtract it from the total. The probability that no birthday's match is (365/365)(364/365)(363/365)(362/365)(361/365)...for how ever many people there are. Then you do 1 - (365/365)(364/365)(363/365)(362/365)(361/365)... to get the probability. For our classroom of 14 (12 students, 1 teacher and 1 TA), the odds of having a match is 22.3%. Just as the odds dictated, we did not have a match. We then found that if you have 23 people, there is a 50% chance of having a match. Having us guess before hand was a very good idea, as it showed us and Dawson if we really knew. It also gave us a chance to think about it logically before learning the way to do it.

After the lecture, we played Jeopardy to review the concepts we learned this week. We were divided into three teams of four. The categories were Matrix Addition, Determents, Combinatorics, Probability, and Presentations. Jeopardy did not go well for our team. We lost a lot of points due to simple mistakes, like giving a Combinatorics answer for a probability question, and adding something instead of subtracting. It was still a good learning experience, because it shows you just how well you know the material. If you don't know it that well, you make all those little mistakes because you try to go fast. We ended up getting 200 points, while the other teams had over 2000 each. At the end of class, Dawson had us fill out a survey on how we feel about the class so far.

All the Origami I made this week!!
There was no study hall today, so we had free time from 3-4. During that time, Ms. Kronenberg and Mr. Mannix came by to visit us. It was nice to see them again, as we won't see Mr. Mannix until July 26th. I have no idea when I will see Ms. Kronenberg again, as she is just spending a few days in Nashville. Ms. Kronenberg had each of us share what our class is like, and what we enjoy about it. From our descriptions, she could tell we are enjoying Vanderbilt, and think the ILC should continue to offer it. After seeing Mr. Mannix and Ms. Kronenberg, we got ready for the Arête showcase. Each class took a turn on the stage, and demonstrated what they did for the past four days. The show was nice, but some classes were more demonstrable then others. (For example, the cricket presentation was just explaining what cricket is, while the Step Team, Fencing and Acting actually presented their work.) I found out from my proctor that next week I have Martial Arts for my Arête class. We shall see how it goes.

After the Arête showcase, we had free time, dinner and more free time. At 7:45, the NEON dance started. Our proctors lead us over to the same room the Arête showcase was in. I do not really enjoy dances. The music is too loud for my taste, and I don't dance because I see no point in it. Unfortunately, we were all required to go to the dance. After about 15 minutes of being on the dance floor, one of the girls who had been trying to get me to dance gave up and asked if I wanted to play cards. I happily obliged, as I would rather do almost anything than be stuck standing there in a room full of loud music and people trying to get me to dance. It turns out she does not really like dances either. We played various card games outside the dance room until the dance ended at 9:30 PM.

At 10:00 PM we had our proctor meeting. Hugh went over the schedule for the weekend. On Saturday, breakfast is at 9, we have class from 10-12, lunch from 12-1, Minute to Win It from 1-3, and a Nashville Sounds baseball game at 5. On Sunday, breakfast is also at 9, and  church services are available. We have SOFT time in the afternoon, and after dinner we have college informations sessions. Unfortunately, these are more general, and not specific to Vanderbilt. I really want to get a presentations about Vanderbilt Admissions, but we'll see what happens. We may be getting college presentations every Sunday evening. We shall see. Tomorrow's going to be an exciting day!!

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