If we let B represent a boy and G represent a girl, here is the sample space:. Suppose we consider the previous example about rolling two dice. Is this unusual? On average, it will occur about 1 in 12 times. Is that unusual enough? We have to be careful when we characterize an event as unusual.
It depends on the context. Suppose we're planning on making a decision one way, unless the probability of a particularly "unusual" event is too high. One example might be the jury in a capital case, punishable by death.
In this example, jurors need to be sure "beyond a reasonable doubt" that the defendant is guilty. If you're that defendant, that's definitely not "unusual" enough! On the other hand, suppose we're planning a picnic on a nice summer day. To find the mean it is easier to just use a table as shown below. Consider the category 7 or more to just be 7. The formula for the mean says to multiply the x value by the P x value, so add a row into the table for this calculation.
Also convert all P x to decimal form. Now add up the new row and you get the answer 2. This means that you expect a household in the U. To find the variance, again it is easier to use a table version than try to just the formula in a line. Looking at the formula, you will notice that the first operation that you should do is to subtract the mean from each x value. Then you square each of these values.
Then you multiply each of these answers by the probability of each x value. Finally you add up all of these values. The numbers in the table above were rounded off because of space limitations, but the answer was calculated using many decimal places. This means that you can expect a U.
Type the x values into L1 and the P x values into L2. Choose Var Stats. This will put 1-Var Stats on the home screen.
If you have the newer operating system on the TI, then your input will be slightly different. The command would be weighted. So for this example, the process would look like:. To find the standard deviation, you would need to program the process into R. So it is easier to just do it using the formula.
What is the expected value in this game? To find the expected value, you need to first create the probability distribution. You also need the probability of winning and losing. Since you are picking a three-digit number, and for each digit there are 10 numbers you can pick with each independent of the others, you can use the multiplication rule. To win, you have to pick the right numbers in the right order.
The first digit, you pick 1 number out of 10, the second digit you pick 1 number out of 10, and the third digit you pick 1 number out of Putting this information into a table will help to calculate the expected value.
Now add the two values together and you have the expected value. Since the expected value is not 0, then this game is not fair. Since you lose money, Arizona makes money, which is why they have the lottery. The reason probability is studied in statistics is to help in making decisions in inferential statistics. To understand how that is done the concept of a rare event is needed.
If, under a given assumption, the probability of a particular observed event is extremely small, then you can conclude that the assumption is probably not correct. An example of this is suppose you roll an assumed fair die times and get a six times, when you should have only rolled a six around times, then you should believe that your assumption about it being a fair die is untrue. Another way to think of this is if the probability of getting such a high value is less than 0.
Another way to think of this is if the probability of getting a value as small as x is less than 0. Why is it "x or more" or "x or less" instead of just "x" when you are determining if an event is unusual? Consider this example: you and your friend go out to lunch every day. Instead of Going Dutch each paying for their own lunch , you decide to flip a coin, and the loser pays for both.
Your friend seems to be winning more often than you'd expect, so you want to determine if this is unusual before you decide to change how you pay for lunch or accuse your friend of cheating. The process for how to calculate these probabilities will be presented in the next section on the binomial distribution.
Data beyond two standard deviations away from the mean is considered " unusual " data. It depends on the context. Suppose we're planning on making a decision one way, unless the probability of a particularly " unusual " event is too high.
A value is " unusual " if it is more than 2 standard deviations away from the mean. An unusual z-score is less than -2 or greater than 2. A z-score of 2 indicates that it is two standard deviations above the mean. What probability is considered unlikely? What is considered an unusual z score? As a general rule, z-scores lower than That is, they are statistically significant outliers.
What happens when z score is too high? So, a high z-score means the data point is many standard deviations away from the mean. What does it mean to be 2 standard deviations away from the mean? What rule do you use to determine if a probability is unusual? If an event is impossible, the probability of the event is 0. If an event is a certainty, the probability of the event is 1.
An unusual event is an event that has a low probability of occurring.
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