Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents. Experiment 2 illustrates the difference between an outcome and an event. A single outcome of this experiment is rolling a 1, or rolling a 2, or rolling a 3, etc. Rolling an even number (2, 4 or 6) is an event, and rolling an odd number (1, 3 or 5) is also an event.

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How likely something is to happen.

1. Title: Reliability analysis of the system lifetime with n working elements and repairing device under condition of the element's fast repair.
2. Probability tells us how often some event will happen after many repeated trials. This topic covers theoretical, experimental, compound probability, permutations, combinations, and more!

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

• heads (H) or
• tails (T)

We say that the probability of the coin landing H is ½

And the probability of the coin landing T is ½

Throwing Dice

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.

The probability of any one of them is 16

Probability

In general:

Probability of an event happening = Number of ways it can happenTotal number of outcomes

Example: the chances of rolling a '4' with a die

Number of ways it can happen: 1 (there is only 1 face with a '4' on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 16

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 45 = 0.8

Probability Line

We can show probability on a Probability Line:

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads.

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Learn more at Probability Index.

Words

Some words have special meaning in Probability:

Experiment: a repeatable procedure with a set of possible results.

Example: Throwing dice

We can throw the dice again and again, so it is repeatable.

The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}

Sample Space: all the possible outcomes of an experiment.

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, etc... }

The Sample Space is made up of Sample Points:

Example: Deck of Cards

• the 5 of Clubs is a sample point
• the King of Hearts is a sample point

'King' is not a sample point. There are 4 Kings, so that is 4 different sample points.

Example: Throwing dice

There are 6 different sample points in the sample space.

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Example Events:

An event can be just one outcome:

• Getting a Tail when tossing a coin
• Rolling a '5'

An event can include more than one outcome:

• Choosing a 'King' from a deck of cards (any of the 4 Kings)
• Rolling an 'even number' (2, 4 or 6)

Hey, let's use those words, so you get used to them:

Example: Alex wants to see how many times a 'double' comes up when throwing 2 dice.

The Sample Space is all possible Outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... {6,3} {6,4} {6,5} {6,6}

The Event Alex is looking for is a 'double', where both dice have the same number. It is made up of these 6 Sample Points:

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

These are Alex's Results:

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After 100 Experiments, Alex has 19 'double' Events ... is that close to what you would expect?

• Early probability
• The rise of statistics
• Statistical theories in the sciences

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Probability and statistics, the branches of mathematics concerned with the laws governing random events, including the collection, analysis, interpretation, and display of numerical data. Probability has its origin in the study of gambling and insurance in the 17th century, and it is now an indispensable tool of both social and natural sciences. Statistics may be said to have its origin in census counts taken thousands of years ago; as a distinct scientific discipline, however, it was developed in the early 19th century as the study of populations, economies, and moral actions and later in that century as the mathematical tool for analyzing such numbers. For technical information on these subjects, seeprobability theory and statistics.

Early probability

Games of chance

The modern mathematics of chance is usually dated to a correspondence between the French mathematicians Pierre de Fermat and Blaise Pascal in 1654. Their inspiration came from a problem about games of chance, proposed by a remarkably philosophical gambler, the chevalier de Méré. De Méré inquired about the proper division of the stakes when a game of chance is interrupted. Suppose two players, A and B, are playing a three-point game, each having wagered 32 pistoles, and are interrupted after A has two points and B has one. How much should each receive?

Fermat and Pascal proposed somewhat different solutions, though they agreed about the numerical answer. Each undertook to define a set of equal or symmetrical cases, then to answer the problem by comparing the number for A with that for B. Fermat, however, gave his answer in terms of the chances, or probabilities. He reasoned that two more games would suffice in any case to determine a victory. There are four possible outcomes, each equally likely in a fair game of chance. A might win twice, AA; or first A then B might win; or B then A; or BB. Of these four sequences, only the last would result in a victory for B. Thus, the odds for A are 3:1, implying a distribution of 48 pistoles for A and 16 pistoles for B.

Pascal thought Fermat’s solution unwieldy, and he proposed to solve the problem not in terms of chances but in terms of the quantity now called “expectation.” Suppose B had already won the next round. In that case, the positions of A and B would be equal, each having won two games, and each would be entitled to 32 pistoles. A should receive his portion in any case. B’s 32, by contrast, depend on the assumption that he had won the first round. This first round can now be treated as a fair game for this stake of 32 pistoles, so that each player has an expectation of 16. Hence A’s lot is 32 + 16, or 48, and B’s is just 16.

Games of chance such as this one provided model problems for the theory of chances during its early period, and indeed they remain staples of the textbooks. A posthumous work of 1665 by Pascal on the “arithmetic triangle” now linked to his name (seebinomial theorem) showed how to calculate numbers of combinations and how to group them to solve elementary gambling problems. Fermat and Pascal were not the first to give mathematical solutions to problems such as these. More than a century earlier, the Italian mathematician, physician, and gambler Girolamo Cardano calculated odds for games of luck by counting up equally probable cases. His little book, however, was not published until 1663, by which time the elements of the theory of chances were already well known to mathematicians in Europe. It will never be known what would have happened had Cardano published in the 1520s. It cannot be assumed that probability theory would have taken off in the 16th century. When it began to flourish, it did so in the context of the “new science” of the 17th-century scientific revolution, when the use of calculation to solve tricky problems had gained a new credibility. Cardano, moreover, had no great faith in his own calculations of gambling odds, since he believed also in luck, particularly in his own. In the Renaissance world of monstrosities, marvels, and similitudes, chance—allied to fate—was not readily naturalized, and sober calculation had its limits.

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