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## Probability Worksheets

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What is probability?

Probability is the branch of mathematics that deals with random events and chance. It is related to our common sense notion of chance (like when we say, "She will probably go to the park.") but uses exact numbers.

Probability is expressed mathematically on a scale from zero (can't happen) to one (will certainly happen). So probability is often written as a percentage or a fraction. For example, a die has six faces, numbered one through six. When the die is thrown, any of the six numbers can show. The chances of any particular number being face up is 1/6 (sometimes read as 1 out of six) - six things could happen (and no others) so if we add up all the possibilities, we should get one. 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1.

It's important to recognize that we don't have to know which number will show up, the numbers are random, but even so, we know the chances of getting each number.

Who invented probability?

Games of chance were first investigated formally by Gerolamo Cardano, an Italian lawyer who was also a wonderful mathematician. He worked on the mathematics of gambling and probability in the 16th century.

 How is probability used in the real world? Mathematical probabilities are all around us. Batting averages, for instance, tell us an estimate of how likely it is that any particular baseball player will get a hit. The numbers are used to decide who is a better batter and how much they are worth when it comes time to trade them or negotiate a new contract. Lotteries are based on probability, and whoever is running the lottery depends on knowing how many tickets they have to sell and the chances someone will win. They use this to decide how much money to pay out (and how much profit they will keep). Earthquakes, sunspots and weather are all random appearing processes, but all can be analyzed using probability. A basic problem in probability. Suppose you have three aces in a five card hand of poker (A,A,A,X,Y - the x and y are other cards). What is the probability that if you discard x and draw a new card, that you will get another ace? Since there are 52 cards in a deck, and you have 5 already, there are a total of 52 - 5 = 47 cards you might draw. Only one of those is an ace. The chances of getting the single ace out of 47 cards total is 1/47. An interesting fact about probability. Sometimes, events we think are very improbable can turn out to be likely when we understand the probability in the mathematical sense. One example is the probability of getting the exact same card dealt simultaneously from two different decks. Take one deck of cards and give a partner another deck. Each of you then turns over one card at a time from the top of your decks. You both continue until all the cards are gone. What are the chances that you will both deal the exact same card at the same time? To count, the cards have to match exactly, same value, same suit. It turns out that the probability of not getting a match is about 36%, which means the chance of getting an exact match is about 64%, so almost 2 out of 3 times, you will get at least one match. This is a very surprising result!