I think to better understand a concept is good to know the motivation behind the creation of the concept. This make the concept 'alive'. A team plays ball such that a total of 60 points is required to win the game, and each inning counts 10 points. The stakes are 24 ducats. By some incident, they cannot finish the game when one side has 50 team A points and the other 30 team B. Pascal starts to write to Fermat about this problem. At first, people argued that the fair way to divide would be for example the aristotle's proportional division: or 5 to 3.
But that didn't seem fair, because team A needed only to win one more game in comparison with team B. None of the solutions seemed fair. So the fair division if the game was not interrupted would be 3 ducats to team B and to team A 21 ducats.
So this opened up the possibility of mathematicians predicting the future. This created industries like insurance and many others. In colloquial language, an average is a single number taken as representative of a list of numbers.
Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an average value. Source: Wikipedia.
So if you are talking about the arithmetic mean, I would say the arithmetic mean is not weighted by the respective probability of an event happen. You just sum all the elements and divide the sum by the total quantity.
This should be the most representative number and which equilibrates your set of numbers. Also, with arithmetic mean we are not talking about the future like expected value. Mean or "Average" and "Expected Value" only differ by their applications, however they both are same conceptually. Since, the average is defined as the sum of all the elements divided by the sum of their frequencies. But for the case of Probability distribution we can't describe a random variable in terms of its frequency beforehand, thus we use the probability instead.
Conceptually, probability of an element is frequency of an event divided by size of sample space. Average on the other hand is used in case where we have the knowledge of frequencies of individual elements and total count of the elements, for example, in case of known data set or sample.
We can simply use the fundamental definition of average to calculate it. Sign up to join this community. The best answers are voted up and rise to the top. If you flip a coin two times, does probability tell you that these flips will result in one heads and one tail? You might toss a fair coin ten times and record nine heads. As you learned in Figure , probability does not describe the short-term results of an experiment.
It gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24, times! He recorded the results of each toss, obtaining heads 12, times. In his experiment, Pearson illustrated the Law of Large Numbers. The Law of Large Numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero the theoretical probability and the relative frequency get closer and closer together.
In other words, after conducting many trials of an experiment, you would expect this average value. The probability that they play zero days is 0. X takes on the values 0, 1, 2.
In this column, you will multiply each x value by its probability. The expected value is 1. The number 1. Calculate the standard deviation of the variable as well. The fourth column of this table will provide the values you need to calculate the standard deviation. For each value x , multiply the square of its deviation by its probability. When all outcomes in the probability distribution are equally likely, these formulas coincide with the mean and standard deviation of the set of possible outcomes.
A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a hour shift. For a random sample of 50 patients, the following information was obtained. What is the expected value? Suppose you play a game of chance in which five numbers are chosen from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A computer randomly selects five numbers from zero to nine with replacement. You pay? Over the long term, what is your expected profit of playing the game?
The values of x are not 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. To win, you must get all five numbers correct, in order. The probability of choosing one correct number is because there are ten numbers. You may choose a number more than once.
The probability of choosing all five numbers correctly and in order is. Since —0. However, each time you play, you either lose?
You are playing a game of chance in which four cards are drawn from a standard deck of 52 cards. You guess the suit of each card before it is drawn. The cards are replaced in the deck on each draw. If you guess the right suit every time, you get your money back and? What is your expected profit of playing the game over the long term? Suppose you play a game with a biased coin. You play each game by tossing the coin once. If you toss a head, you pay?
If you toss a tail, you win? If you play this game many times, will you come out ahead? Add the last column of the table.
You lose, on average, about 67 cents each time you play the game so you do not come out ahead. Suppose you play a game with a spinner. You play each game by spinning the spinner once. If you land on red, you pay? If you land on green, you win? Complete the following expected value table. Like data, probability distributions have standard deviations. Add the last column in the table. The standard deviation is the square root of 0. Toss a fair, six-sided die twice. Tossing one fair six-sided die twice has the same sample space as tossing two fair six-sided dice.
The sample space has 36 outcomes:. Use this value to complete the fourth column. On May 11, at PM, the probability that moderate seismic activity one moderate earthquake would occur in the next 48 hours in Iran was about Suppose you make a bet that a moderate earthquake will occur in Iran during this period.
If you win the bet, you win? If you lose the bet, you pay? If you bet many times, will you come out ahead? Explain your answer in a complete sentence using numbers. What is the standard deviation of X? Construct a table similar to Figure and Figure to help you answer these questions.
If you make this bet many times under the same conditions, your long term outcome will be an average loss of? On May 11, at PM, the probability that moderate seismic activity one moderate earthquake would occur in the next 48 hours in Japan was about 1.
As in Figure , you bet that a moderate earthquake will occur in Japan during this period. Find the mean and standard deviation of X. Some of the more common discrete probability functions are binomial, geometric, hypergeometric, and Poisson. Sign In Email: Password: Forgot password? Video Available. Note that this makes sense intuitively. Solution Before doing the math, we suggest that you try to guess what the expected value would be. It might be a good idea to think about the examples where the Poisson distribution is used.
Solution We provide two ways to solve this problem.
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