Do Now If you were creating your own table of random digits, which of the following strings of numbers would you be more likely to get? 8327651294 0123456789 Answer: Both are equally likely! Why?

7.1: Thinking about Chance

The Idea of Probability Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. Definition: The probability of any outcome of a chance process is a number between 0 (never occurs) and 1 (always occurs) that describes the proportion of times the outcome would occur in a very long series of repetitions.

Randomness and Probability We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

Examples 1.) The table of random digits was produced by a random mechanism that gives each digit probability 0.1 of being a 0. What proportion of the first 200 digits in the table are 0s? 21/200 or .105 2.) Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given. 0 a.) This event is impossible and will never occur 0.01 0.3 b.) This event is certain and will always occur 0.6 0.99 c.) This event is very unlikely 1 d.) This event will occur more often than not

Myths about Chance Behavior Example Lindsay is no mathematician but thinks she has just discovered a foolproof system to get rich playing roulette. She has been observing the spin of the wheel at the casino for several days. During this time she had noticed that it was surprisingly normal for there to be a sequence of spins when the ball fell into only black or only red slots. But five in a row of the same color was very unusual and six only occurred a few times a day.

This was going to be her system This was going to be her system. The chances of the ball falling into a slot of the same color six times in a row were tiny. So, she would watch, and once it fell into the same color 5 times in a row, she would bet big money that the next one was the other color. She was bound to win more times than she lost because six in a row was so rare. She was so confident that she had already begun to think about how she would spend the money! What is wrong with Lindsay’s system for getting rich?

Myth 1: The Law of Averages Lindsay believed the "law of averages" meant black had to come up to balance out the reds.   This is simply not the case, items of chance such as roulette  wheels, dice, and coins do not have any memories. They do not  know what they did before, and in short instances improbable  things may happen, when compared to the long run.

As said earlier, the idea of probability is that randomness isregular  in the long  run.  Unfortunately, our intuition about randomness tries to tell us that random  phenomena should also be regular in the short run. Try this:  Flip a coin 100 times.  Record the results using a T for tails and an H for heads. Group results in 5s so it’s easier to keep track. Count how many “runs” of 2 or more you got.

Myth 2: Short-term vs. Long-term regularity "She's on Fire!!!" If a basketball player makes several consecutive shots, both the fans and her teammates believe that she is "on fire!" and is more likely to make the next shot.This is wrong.  Studies show that players perform consistently, not in streaks.   If a player makes half her shots in the long run, her hits and misses behave just  like the tosses of a coin ­ which means runs of hits and misses are more common than our intuition expects. Conclusion Random phenomena behave regularly in the long term, but not in the short term.

Myth 3: The Surprising Coincidence Julie is spending the summer in London. One day, on the third floor of the Victoria and Albert Museum, she runs into Jim, a casual friend from college.  "OMG, how unusual!   Maybe we were fated to meet!" Well, maybe not.  Although it is unlikely for Julie to run into a particular  acquaintance that day, it is not at all unlikely that  she would run into some acquaintance.  After all, a typical adulthas about 1500 casual acquaintances.  When something unusualhappens, we look back and say "Wasn't that unlikely?"  We  would have said the same thing if any of the other 1500  unlikely events had happened.

EXERCISES 1.  An unenlightened gambler a.  Coach Wensel knows that red and black are  equally likely to occur on  each spin of a roulette  wheel.  He observes 6 consecutive reds occur and  bets heavily on black at the next spin. Asked why,  he explains that black is "due  because of the law  of averages."  Explain to this man what is wrong  with this reasoning. Myth of law of averages

b.  After listening to your eloquent argument, Coach Wensel moves on to a poker game.  He is dealt 5 straight red cards.  He remembers what you said and  assumes that the next carddealt in the same hand is equally like to be red or black.  Is  Coach Wensel right or wrong, and most importantly, why? No, there are more black cards left. What happened before affects what happens after

2.)The basketball player Dwight Howard misses about  half of his free throws over an entire season. In today'sgame, Howard is successful on his first three free  throws.  The TV commentator says, “Howard's techniquereally looks good today."  Explain why the claim that  Dwight Howard has improved his free throw shooting  technique is not justified. myth of short run regularity

Personal Probabilities What’s the probability that the Giants will win next year’s Superbowl? Note that we can’t base this answer on the long-term pattern in many repetitions. Instead, we base the answer on our personal judgment. A personal probability is a number between 0 and 1 that expresses someone’s personal judgment about how likely an outcome is.

Personal Probabilities (Continued) Often important decisions are based on a personal probability as opposed to a long-term proportion. Example 1: Should I bet on South Carolina to beat Clemson in the upcoming game? Example 2: Should my company set up offices in Manhattan? Example 3: Should I take the bus downtown instead of trying to drive and find a parking space?

All of these are situations where the outcome of interest comes from a one-time event, not from repeatable trials. Which probability is based on a one-time event rather than repeatable trials? The probability of rolling a “6” with a fair die. The probability of tossing a dime and a penny and getting two “heads”. The probability of rain next Wednesday. The probability of making a 3-point shot in basketball.

Dealing with Very Small Probabilities It’s hard for us to comprehend the magnitude of very small probabilities. This makes it difficult for us to assess the risks of rare events. Example 1: What is the probability of a terrorist hijacking a commercial airplane? Example 2: What is the probability of our house being destroyed by a tornado?

Dealing with Very Small Probabilities (continued) Important decisions must be made based on the risks of these rare events. Example 1: Should airports spend the money & manpower to search passengers for weapons & bomb materials? Example 2: Should we purchase tornado insurance (fire, hurricane, etc.?) Probabilities associated with rare events cannot be easily assessed. Experts use complicated probability models to try to estimate these probabilities.

Dealing with Very Small Probabilities (continued more) Psychologically, we tend to worry less about events we encounter often and which we feel we have control over (example: driving in a car) We tend to worry more about events we encounter rarely and of which we lack knowledge or control (examples: air travel, natural disasters, asbestos) Example: Which is riskier, a cross-country airline flight, or driving to the airport to catch the flight? Driving to the airport

Interesting Example: If a baby is sleeping at home, would a parent leave the baby alone to drive off on a 15-minute errand? Which option poses a greater risk to the baby’s well-being?

7.1 Practice Complete pg. 418/7.8-7.10 pg. 421-422/7.11,7.12, 7.14-7.16, 7.18