Counting and Probability Lecture 42 Section 9.1 Robb T. Koether Hampden-Sydney College Wed, Apr 9, 2014 Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 1 / 17
1 Probability 2 The Monty Hall Problem 3 The Incredible Dice Problem 4 Assignment Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 2 / 17
Outline 1 Probability 2 The Monty Hall Problem 3 The Incredible Dice Problem 4 Assignment Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 3 / 17
Probability Definition (Probability) The probability of an event is the fraction of the time that it occurs in the long run. The phrase in the long run means infinitely long. Any amount of experimentation will produce only an approximation to the true probability, although the more experiments we perform, the more reliable the estimate. Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 4 / 17
Experiments Definition (Experiment, Sample space) An experiment is a procedure that, when followed, leads to an outcome. We observe a characteristic of the outcome, which is a variable. The sample space of the experiment is the set of all possible values of the variable. If the experiment is to toss a coin and the variable is whether the coin landed heads or tails, then the sample space is {H, T }. If the experiment is to roll a die and the variable is the number that turns up, then the sample space is {1, 2, 3, 4, 5, 6}. Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 5 / 17
Calculating Probability If the sample space is finite, with n members, and the members are equally likely, then the probability of each member is 1 n. The probability of heads in a coin toss is 1 2. The probability of 3 in a die roll is 1 6. Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 6 / 17
Events Definition (Event) An event is a subset of the sample space. If the outcome of the experiment is in the event, we say that the event occurred. Otherwise, it did not occur. In the die rolling experiment, the event {1, 3, 5} occurs if the number is odd. Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 7 / 17
Probability of an Event If all the outcomes in the sample space S are equally likely, then the probability of an event E is P(E) = E S. When drawing a card from a deck of 52 cards, what is the probability of drawing an ace or a red card? When rolling two dice, what is the probability of getting a total of 6? When tossing three coins, what is the probability of getting exactly 2 heads? Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 8 / 17
Outline 1 Probability 2 The Monty Hall Problem 3 The Incredible Dice Problem 4 Assignment Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 9 / 17
The Monty Hall Problem There are three doors: Door 1, Door 2, Door 3. Behind one door is a new Ferrari. Behind the other two doors there are goats. You select one of the doors. If the game show host opens that door, you get whatever is behind it. You select Door 1. The host opens one of the other two doors and shows you a goat and asks you if you want to switch to the remaining door. Should you switch? Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 10 / 17
The Monty Hall Problem Ferrari Ferrari Ferrari 1 2 3 1 2 3 1 2 3 Your choice Host opens door The other door If you switch, what are your chances of winning? Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 11 / 17
The Monty Hall Problem Ferrari Ferrari Ferrari 1 2 3 1 2 3 1 2 3 Your choice Host opens door The other door If you stay, your chances are 1 3 of winning the Ferrari. Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 11 / 17
The Monty Hall Problem Ferrari Ferrari Ferrari 1 2 3 1 2 3 1 2 3 Your choice Host opens door The other door If you switch, your chances are 2 3 of winning the Ferrari. Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 11 / 17
Outline 1 Probability 2 The Monty Hall Problem 3 The Incredible Dice Problem 4 Assignment Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 12 / 17
The Incredible Dice Problem There are three dice: Die A, Die B, Die C. They contain the following numbers: Die A: 1, 6, 8, 10, 15, 17 Die B: 2, 4, 9, 11, 13, 18 Die C: 3, 5, 7, 12, 14, 16 There are 2 players and each chooses a die. They roll their dice and the high number wins. Which die would you choose? Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 13 / 17
The Incredible Dice Problem 1 Die B 2 4 9 11 13 18 Die A 6 8 10 15 A wins B wins 17 Die A beats Die B only 17 36 < 1 2 of the time. Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 14 / 17
The Incredible Dice Problem 2 Die C 3 5 7 12 14 16 Die B 4 9 11 13 B wins C wins 18 Die B beats Die C only 17 36 < 1 2 of the time. Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 14 / 17
The Incredible Dice Problem 3 Die A 1 6 8 10 15 17 Die C 5 7 12 14 C wins A wins 16 Die C beats Die A only 17 36 < 1 2 of the time. Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 14 / 17
Outline 1 Probability 2 The Monty Hall Problem 3 The Incredible Dice Problem 4 Assignment Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 15 / 17
Collected Collected Sec. 8.1: 11, 18. Sec. 8.2: 19, 52. Sec. 8.3: 10, 42. Sec. 8.5: 7, 11. Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 16 / 17
Assignment Assignment Read Sections 9.1, pages 516-522. Exercises 1, 3, 4, 8, 11, 12, 13, 15, 18, 19, 20, page 523. Robb T. Koether (Hampden-Sydney College) Counting and Probability Wed, Apr 9, 2014 17 / 17