Probability. 13 February Math 210G 13 February /21

Transcription

1 Probability 13 February 2012 Math 210G 13 February /21

2 Homework Assignment (forgot to mention last time) Assignment 3 is on the course website. Since I forgot to mention it on Friday I m pushing the due date to next Monday. The assignment is about Tammy and Rex encryption. While the problems can be done by hand, they were written with the intention of using the TammyAndRex spreadsheet, which is available from the Handouts link of the course website. I strongly encourage you to use it. I ll be happy to answer questions about using it. Math 210G 13 February /21

3 Now to This Week s Topic The study of probability dates back to the mid 17th century through correspondence between two mathematicians, Pierre de Fermat and Blaise Pascal. Math 210G 13 February /21

4 Here is a quote from Calculus, Volume II by Tom M. Apostol (2nd edition, John Wiley & Sons, 1969): A gambler s dispute in 1654 led to the creation of a mathematical theory of probability by two famous French mathematicians, Blaise Pascal and Pierre de Fermat. Antoine Gombaud, Chevalier de Méré, a French nobleman with an interest in gaming and gambling questions, called Pascal s attention to an apparent contradiction concerning a popular dice game.the game consisted in throwing a pair of dice 24 times; the problem was to decide whether or not to bet even money on the occurrence of at least one double six during the 24 throws. Math 210G 13 February /21

5 A seemingly well-established gambling rule led de Méré to believe that betting on a double six in 24 throws would be profitable, but his own calculations indicated just the opposite. This problem and others posed by de Méré led to an exchange of letters between Pascal and Fermat in which the fundamental principles of probability theory were formulated for the first time. Although a few special problems on games of chance had been solved by some Italian mathematicians in the 15th and 16th centuries, no general theory was developed before this famous correspondence. Math 210G 13 February /21

6 Simulation of de Méré s Problem Roll a pair of dice 24 times. If you roll double six at least once, then you win. If you never get double six on any of the 24 rolls, you lose. Once you are done, submit A Win B Lose We will determine what percentage of the class wins. Math 210G 13 February /21

7 It turns out that the probability of losing is (35/36) 24 which is approximately.51, or about 51%. Math 210G 13 February /21

8 It turns out that the probability of losing is (35/36) 24 which is approximately.51, or about 51%. Then the probability of winning is approximately.49, or slightly less than half. Thus, de Méré was right that it is a little less than even money to bet on getting a double six in 24 rolls. Later on we will see how to come up with this calculation. Math 210G 13 February /21

9 It turns out that the probability of losing is (35/36) 24 which is approximately.51, or about 51%. Then the probability of winning is approximately.49, or slightly less than half. Thus, de Méré was right that it is a little less than even money to bet on getting a double six in 24 rolls. Later on we will see how to come up with this calculation. Doing the computation above requires a scientific calculator. There are free websites that do the same thing. One is Math 210G 13 February /21

10 The reason I m mentioning calculators now is that there will be times that we will need some sort of calculator. In particular, when we discuss interest rates near the end of the semester, we ll need to do some fairly complex computations. I don t encourage you to buy a scientific calculator. Only get one if you think you ll have use of it outside this class. There are plenty of websites that do calculations, including financial calculations. The calculator program on a computer will do the same thing, so you ll be able to do any calculation we ll need Math 210G 13 February /21

11 If you have a smart phone you may already have a scientific calculator. For example, the iphone comes with a calculator app. Rotating the phone toggles between a regular and scientific calculator. Math 210G 13 February /21

12 Simulating de Méré s Problem with Excel We will use the Microsoft Excel spreadsheet demereproblem.xlsx to simulate our experiment. The advantage is that we can conduct many trials in a short amount of time; many more than we could do by actually rolling dice. This spreadsheet, and all others we will use, will be on the course website. Math 210G 13 February /21

13 Theoretical and Experimental Probability This week we will explore some of the ideas that are used in this calculation. But first, we will focus on getting a better understanding of what is the meaning of probability. Math 210G 13 February /21

14 Theoretical and Experimental Probability This week we will explore some of the ideas that are used in this calculation. But first, we will focus on getting a better understanding of what is the meaning of probability. If you conduct an experiment, such as the dice rolling we did earlier, you can compute an experimental probability, as we did. Our percentage of wins for the entire class is an experimental probability for winning that game. Math 210G 13 February /21

15 If we say the probability of getting heads when you flip a coin is 50%, this is a theoretical probability. It is telling us what we expect to get if we flip a bunch of coins. Math 210G 13 February /21

16 If we say the probability of getting heads when you flip a coin is 50%, this is a theoretical probability. It is telling us what we expect to get if we flip a bunch of coins. The reality is a little more complicated, as we ll explore. Roughly, to say that the probability (or chance) of flipping a coin and getting heads is 50%, or.5, then on average, if you flip a coin many times, you will expect 50% of the flips being heads. However, what happens in a given set of flips can be nearly anything. Math 210G 13 February /21

17 The theoretical probability allows us to estimate what will happen when we conduct an experiment. However, we can use an experimental probability to estimate theoretical probability. Therefore, each can be used to estimate the other. Math 210G 13 February /21

18 The theoretical probability allows us to estimate what will happen when we conduct an experiment. However, we can use an experimental probability to estimate theoretical probability. Therefore, each can be used to estimate the other. For example, casinos can estimate how much money they will make from a game by knowing the theoretical probability of winning that game. What actually happens on a given day is an experimental probability. Because they have so many people playing, their estimates are usually pretty good. Math 210G 13 February /21

19 The theoretical probability allows us to estimate what will happen when we conduct an experiment. However, we can use an experimental probability to estimate theoretical probability. Therefore, each can be used to estimate the other. For example, casinos can estimate how much money they will make from a game by knowing the theoretical probability of winning that game. What actually happens on a given day is an experimental probability. Because they have so many people playing, their estimates are usually pretty good. There is always a chance that somebody wins big on a given day. But, if they estimate income for longer periods of time, they ll be even more accurate. Math 210G 13 February /21

20 A Coin Flip Experiment Flip a coin once and record the number of heads (either 0 or 1) with your clicker. We will show the class data. Math 210G 13 February /21

21 Another Coin Flip Experiment Flip a coin 20 times and enter the number of heads you got. We will tabulate the data for the entire class. What does the data indicate to you about the probability of getting a heads on a flip of a coin? Math 210G 13 February /21

22 Simulating the Coin Flip Experiment with Excel We will use the spreadsheet Coin Flip.xlsx to simulate flipping a coin. Math 210G 13 February /21

23 Simulating the Coin Flip Experiment with Excel We will use the spreadsheet Coin Flip.xlsx to simulate flipping a coin. We will simulate flipping different numbers of coins with the spreadsheet. One thing to think about is how do you think the number of coins we flip will affect the results. How did flipping one coin versus 20 coins affect our results in the previous experiment? Math 210G 13 February /21

24 Observations from the Simulation How did the variability of the percentage of heads change as we increased the number of flips? Math 210G 13 February /21

25 Observations from the Simulation How did the variability of the percentage of heads change as we increased the number of flips? The percentage of heads didn t change as much from trial to trial when we flipped more coins. We are therefore getting a better estimate of the theoretical probability the more coins we flip. Math 210G 13 February /21

26 Observations from the Simulation How did the variability of the percentage of heads change as we increased the number of flips? The percentage of heads didn t change as much from trial to trial when we flipped more coins. We are therefore getting a better estimate of the theoretical probability the more coins we flip. Do you think the spreadsheet confirmed that the probability is 50% to get a heads (assuming the spreadsheet does correctly simulate flipping a coin)? Math 210G 13 February /21

27 Quiz Question If you want to know the probability of flipping a coin and getting heads, which experiment is most likely to give you the most accurate estimate of the probability? A Flipping a coin once B Flipping a coin 100 times C Flipping a coin 10,000 times Math 210G 13 February /21