The probability of getting a given number of heads from four flips is, then, simply the number of ways that number of heads can occur, divided by the number of total results of four flips, We can then tabulate the probabilities as follows.
Since we are absolutely certain the number of heads we get in four flips is going to be between zero and four, the probabilities of the different numbers of heads should add up to 1. Summing the probabilities in the table confirms this. Further, we can calculate the probability of any collection of results by adding the individual probabilities of each. Suppose we'd like to know the probability of getting fewer than three heads from four flips.
There are three ways this can happen: zero, one, or two heads. So to calculate the probability of one outcome or another, sum the probabilities. To get probability of one result and another from two separate experiments, multiply the individual probabilities.
What's the probability of getting one head in each of two successive sets of four flips? The probability for any number of heads x in any number of flips n is thus:.
But there's no need to sum the combinations in the denominator, since the number of possible results is simply two raised to the power of the number of flips.
So, we can simplify the expression for the probability to:. Let's see how the probability behaves as we make more and more flips. Since we have a general formula for calculating the probability for any number of heads in any number of flips, we can graph of the probability for various numbers of flips. In every case, the peak probability is at half the number of flips and declines on both sides, more steeply as the number of flips increases.
This is the simple consequence of there being many more possible ways for results close to half heads and tails to occur than ways that result in a substantial majority of heads or tails. The RPKP experiments involve a sequence of random bits, in which the most probable results form a narrow curve centred at A document giving probabilities for results of bit experiments with chance expectations greater than one in thousand million runs is available , as is a much larger table listing probabilities for all possible results.
The latter document is more than K bytes and will take a while to download, and contains a very large table which some Web browsers, particularly on machines with limited memory, may not display properly.
The normal distribution gives the probability for x heads in n flips as:. To show how closely the probability chart approaches the normal distribution even for a relatively small number of flips, here's the normal distribution plotted in red, with the actual probabilities for number of heads in flips shown as blue bars.
The probability the outcome of an experiment with a sufficiently large number of trials is due to chance can be calculated directly from the result, and the mean and standard deviation for the number of trials in the experiment. For additional details, including an interactive probability calculator, please visit the z Score Probability Calculator. This is all very persuasive, you might say, and the formulas are suitably intimidating, but does the real world actually behave this way?
Well, as a matter of fact, it does, as we can see from a simple experiment. Get a coin, flip it 32 times, and write down the number of times heads came up. Now repeat the experiment fifty thousand times. When you're done, make a graph of the number of flip sets which resulted in a given number of heads. Hmmmm…32 times 50, is 1. Instead of marathon coin-flipping, let's use the same HotBits hardware random number generator our experiments employ. It's a simple matter of programming to withdraw 1.
The results from this experiment are presented in the following graph. The red curve is the number of runs expected to result in each value of heads, which is simply the probability of that number of heads multiplied by the total number of experimental runs, 50, The blue diamonds are the actual number of 32 bit sets observed to contain each number of one bits.
It is evident that the experimental results closely match the expectation from probability. Just as the probability curve approaches the normal distribution for large numbers of runs, experimental results from a truly random source will inexorably converge on the predictions of probability as the number of runs increases. If your Web browser supports Java applets, our Probability Pipe Organ lets you run interactive experiments which demonstrate how the results from random data approach the normal curve expectation as the number of experiments grows large.
Performing an experiment amounts to asking the Universe a question. For the answer, the experimental results, to be of any use, you have to be absolutely sure you've phrased the question correctly. When searching for elusive effects among a sea of random events by statistical means, whether in particle physics or parapsychology, one must take care to apply statistics properly to the events being studied. Misinterpreting genuine experimental results yields errors just as serious as those due to faults in the design of the experiment.
Evidence for the existence of a phenomenon must be significant , persistent , and consistent. Statistical analysis can never entirely rule out the possibility that the results of an experiment were entirely due to chance—it can only calculate the probability of occurrence by chance.
Only as more and more experiments are performed, which reproduce the supposed effect and, by doing so, further decrease the probability of chance, does the evidence for the effect become persuasive. To show how essential it is to ask the right question, consider an experiment in which the subject attempts to influence a device which generates random digits from 0 to 9 so that more nines are generated than expected by chance.
Each experiment involves generation of one thousand random digits. We run the first experiment and get the following result:.
Ask them to share ways that people can find current information about arctic ice. Their list will probably include television, radio, newspapers, internet, etc. Tell them that they will search the internet for headlines about the melting of arctic ice. Brainstorm some of the search strings they might type into a search engine such as Google.
Some options are "melting arctic ice', "arctic sea ice", "arctic ice". Remind students to look at where the headline is from. Students will find a headline and write it down along with the source of the headline. Then they will read the associated article or information, making note of 2 or 3 main points of the article. Allow 15 or 20 minutes for this activity. Then share the headlines and points they students have noted. You may want to record the information on chart paper or the board. The resulting chart will likely reflect different points of view regarding the melting of sea ice and will provide an opportunity for a class discussion regarding the uncertainties in science.
Focus Questions: How is arctic sea ice changing over time? Distribute the arctic sea ice data handout to students. This data shows sea ice extent, but does not provide information about ice thickness.
Where do you think this data comes from? This helps explain whay data does not exist before , since reliable data did not exist before then. If you were a scientist and given the task of sharing the news of this data with the world, how would you present it? What would you expect to see if you picked one year and tracked the sea ice in the arctic? Have students make a prediction in their science notebooks, then graph one year either by hand or using Excel. Students can choose a line graph, bar chart, or possibly a pictogram.
After their graphs are complete, ask them to respond to the following question in their science notebooks:. If the ice is fluctuating yearly, how would we go about trying to communicate what our data shows about sea ice over many years?
Show the scatterplot example to students, and depending on student level, review how to make a scatterplot using one month of the year. Individually or in pairs, students choose a month of the year to track sea ice extent. Students can create a scatterplot graph either by hand or by using Excel. Have students describe the overall trend of the data in their science notebooks, and make a trend line line of best fit for their graphs.
Ask students to respond to the following questions in their science notebooks: What is going on with respect to sea ice extent in the years from to present, as shown by the data? Why do scientists use graphs instead of data tables to present large amounts of data? Have students share their scatterplots with the class and discuss their answers to the questions with the whole group.
Show students the sea ice animation. Note that the years of the animation correspond very closely to the years they graphed. Ask students what they noticed. Was the ice ever similar to September during the animation? Why do they think the sea ice extent in is less than any of the previous years in the animation? The guillemots are ice-dependent arctic nesting seabirds. A year study has documented changes in their habitat related to a retreating ice pack, and changes in their success in reproduction related to the distance they have to fly to find food, as well as the movement northward of horned puffins, a subarctic species.
This video is a good introduction to the concept that physical changes in climate and ice and have effects on specific animals and their ecological relationships. If time, space, and materials are not available, this activity can be a demonstration by the teacher.
Instructions: 1. Pour warm water into the cup, so that the cup is about half full. Add an ice cube to the cup and carefully mark the level of the water with a permanent marker or a piece of tape.
Note : Federal employees and applicants for federal employment have a different complaint process. Within 10 days of the filing date of your charge, we will send a notice of the charge to the employer. In some cases, we will ask both you and the employer to take part in our mediation program.
If the laws the EEOC enforces do not apply to your claims or if your charge is untimely, or we decide that we probably will not be able to determine if the law was violated, we will close the investigation of your charge and notify you.
If you and the employer agree to mediation , a mediator will try to help you both reach a voluntary settlement. Mediation allows you and the employer to talk about your concerns. Mediators don't decide who is right or wrong, but they are very good at suggesting ways to solve problems and disagreements. If the charge is not sent to mediation, or if mediation doesn't resolve the charge, we usually ask the employer to give us a written answer to your charge called "Respondent's Position Statement".
You will receive an email once we receive the position statement and it is available for you to review. Log in to the Public Portal to obtain a copy of the position statement. We ask that you provide a response within 20 days from the date you receive it. We may also ask the employer to answer questions we have about the claims in your charge.
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