5.1 Day 1

IntroThe Idea of Probability

Myths about Randomness

Is the claim 1 in 6 wins legit?

• Dr. Pepper is running a new promotion. Their label states that 1 in 6 wins. The winning caps say “You’re a winner!” and the remaining caps say “Try Again”

• You and six other friends go to Dillons. Each of you buys a Dr. Pepper. Three of you are winners!

• How likely is this? Is Dr. Peppers claim legit?

Let’s see…

• 1 – 5 represent try again• 6 – represents You’re a winner

• Roll your die seven times to imitate the process of seven friends buying their sodas. How many of them won a prize?

• Repeat four more times. • In your five repetitions, how many times did three or more of the

group win a prize?• Combining all our results, what percent of the time did the friends

come away with three or more prizes.?• Does it is seem plausible that the company is telling the truth, but

that the seven friends are just lucky?

Randomness, Probability and Simulation

• http://bcs.whfreeman.com/tps4e/#628644__666397__Probability applet

Coin Toss

• Using the website applet– Click show true probability

• Toss a coin 10 times – What is the probability of getting heads?• Click reset

• Toss it gain 10 more times– Now what is the probability?– Keep repeating this process, what do you notice?

Toss a coin 100 times

• Reset the applet• Toss 40, then 40 more, and then 20 more. (it

can only go to 40 so we have to make 100)• Is the probability .5?• Keep tossing more an more without hitting

reset.• What happens to the probability?

Law of Large Numbers

• As the number of trials increases, the experimental probability will approach the theoretical probability.

• A "law of large numbers" is the idea that as the number of trials of a random process increases, the percentage difference between the expected and actual values goes to zero.

Probability

• The probability of a chance process is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions.

Likelihood of an event to occur

0 will never occur1 will occur every repetition

.5 likely to occur half the time.

Remember probability is never exact, just because something happens 60% of the time once, doesn’t mean it will always occur 60% of the time.

What is the likelihood of a jar lid landing face down?

• Is it .5?

Example

• Example: Why is male life insurance more than female life insurance?

• National Center for Health Statistics says that the proportion of men aged 20 to 24 who die in any one year is 0.0015. For women that age, the probability of death is about 0.0005.

• Therefore a man’s policy aged 20 – 24 is 3 times more expensive than a females.

Myths about Randomness• Without actually flipping a coin, imagine the first

toss. Write down the result, H or T. Repeat 50 times. Separate every 5th toss by a line like HTTHH|HTTHHT ….

• Count the length of each run, like TT = 2, HHH = 3, TT = 2, HH = 2 …

• Repeat the process using the randint function for a total of 50 simulated tosses. Keep track of each result.

• Did you or your calculator have the longest run?Run Length 2 3 4 5 6 7 8 9 10Frequency

Frequency using calc.

Myths about Randomness

• The myth of short term regularity – The idea of probability is that randomness is predictable in the

long term– Streaks of the same outcome are often viewed with suspicion

even though they occur quite often just by chance• The myth of the law of averages

– People mistakenly believe that outcomes will even out in a short number of trials.

– For example, if you flip four heads in a row, is the next flip more likely to be tails? No!

– They will even out in the long run. In a trial of 10,000 tosses will the first four flips mean anything?

Example:

• If a basketball player makes several consecutive shots both the fans and teammates believe that the player is more likely to make the next shot. This is wrong! Careful study has shown that runs of baskets make or missed arm o more frequent in basketball than would be expected if each shot is independent of the player’s previous shots. If a player makes half the shots in the long run, than each shot, has a 50% chance of being made, regardless of the short run.

Law of Averages

• Myth! – Say you toss a coin a 5 times for outcome of TTTTT, the next toss must be a head right? False! The outcome of the last five tosses have no effect on the outcome of the next. In the long run, the outcomes will even out, but the short run is much less predictable.

PRACTICE PROBLEMS

According to the “Book of Odds,” the probability that a randomly selected US adult usually etas breakfast is 0.61.

Explain what probability 0.61 means in this setting

Why doesn’t this probability say that if 100 US adults are chosen at random, exactly 61 of them usually eat breakfast?

Solution!

• A) This means that if you asked a large sample of US adults whether they usually eat breakfast, about 61% of them would answer yes.

• B) If a random sample of 100 adults, we would expect that around 61 of them will usually eat breakfast. However the exact number will vary from sample to sample.

Practice continued• Probability is a measure of how likely an outcome is to

occur. Match one of the probabilities that follow with each statement. Be prepared to defend your answer.

0 0.01 0.3 0.6 0.99 1• A) The outcome is impossible. It can never occur• B) The outcome is certain. It will occur on every trial.• C) The outcome is very unlikely, but it will occur once in a

while in a long sequence of trials.• D) The outcome will occur more often than not.

Page 292 1, 3,7,9,11

• Sometimes the police us a lie detector to help determine whether a suspect is telling the truth. A lie detector isn’t foolproof – sometimes it suggests that a person is lying when they’re actually telling the truth. Other times, the test says that the suspect is being truthful when the person is actually lying. For one brand of polygraph machine, the probability of a false positive is 0.08.

• Interpret this probability as a long-run relative frequency.

• Which is more serious in this case: a false positive or a false negative? Justify your answer.

• Suppose a married man and woman both carry a gene for cystic fibrosis but don’t have the disease themselves. According to the laws of genetics, the probability that their first child will develop cystic fibrosis is 0.25.

• • Explain what this probability means.• • • Whey doesn’t this probability say that if the couple has 4

children, one of them is guaranteed to get cystic fibrosis.

• 7. The figure below shows the results of a basketball player shooting several free throws. Explain what this graph says about change behavior in the short run and long run.

• 9. Due for a hit. A very good professional baseball player gets a hit about 35% of the time over an entire season. After the play failed to hit safely in six straight at bats, a TV commentator said, “He is due for a hit by the law of the averages.” Is that right? Why?

• Playing Pick 4. The pick 4 games in many state lotteries announce a four digit winning number each day. You can think of the winning number as a four digit group from a table of random digits. You win the jackpot if your choice matches the winning number. The winnings are divided among all players who matched the winning number. That suggests a way to get an edge.

• a) The winning number might be, for example, either 2873 or 9999. Explain why these two outcomes have the same probability.

•

• b) If you asked many people whether 2873 or 9999 is more likely to be the randomly chosen winning number, most would favor one of them. Use the information in this section to say which one and to explain why. How might this affect the four digit number you would choose?

Assignment

• P. 294 1,3,7,9,11