We also know the first card was an ace, therefore:. As an example, consider the experiment of rolling a die and flipping a coin. To say that two events are independent means that the occurrence of one does not affect the probability of the other. In probability theory, to say that two events are independent means that the occurrence of one does not affect the probability that the other will occur.
The concept of independence extends to dealing with collections of more than two events. To show that two events are independent, you must show only one of the conditions listed above. If any one of these conditions is true, then all of them are true.
Translating the symbols into words, the first two mathematical statements listed above say that the probability for the event with the condition is the same as the probability for the event without the condition.
For independent events, the condition does not change the probability for the event. As an example, imagine you select two cards consecutively from a complete deck of playing cards. The two selections are not independent. The result of the first selection changes the remaining deck and affects the probabilities for the second selection.
Because the deck of cards is complete for both selections, the first selection does not affect the probability of the second selection. When selecting cards with replacement, the selections are independent. Independent Events : Selecting two cards from a deck by first selecting one, then replacing it in the deck before selecting a second is an example of independent events.
Consider a fair die role, which provides another example of independent events. If a person roles two die, the outcome of the first roll does not change the probability for the outcome of the second roll. Two friends are playing billiards, and decide to flip a coin to determine who will play first during each round. For the first two rounds, the coin lands on heads. They decide to play a third round, and flip the coin again. What is the probability that the coin will land on heads again? First, note that each coin flip is an independent event.
The side that a coin lands on does not depend on what occurred previously. Also recall that the following statement holds true for any two independent events A and B:. Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.
Combinatorial techniques are applicable to many areas of mathematics, and a knowledge of combinatorics is necessary to build a solid command of statistics. It involves the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties. Aspects of combinatorics include: counting the structures of a given kind and size, deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria.
Several useful combinatorial rules or combinatorial principles are commonly recognized and used. Each of these principles is used for a specific purpose. But what if the events overlap — if there are outcomes that are part of both events? But that's a different matter. Here is an answer I gave about this in , not in the context of probability, but in an elementary counting question: The Difference between And and Or My son had a question that was marked wrong on his paper.
He pointed out to me that by the way it was worded, he felt as though he were correct. Here is the question: There are 3 knives, 4 spoons, 4 forks. What fraction of the utensils are spoons OR forks? I understand the way he read it to be OR meaning one or the other. Nothing is both a spoon and a fork! At least not in this problem. So "and" would have been inappropriate. There are no utensils that are spoons and forks. This is where the confusion and ambiguity come in! There are 8 utensils that are spoons or forks.
Your son read it in a way that is commonly used in nontechnical English, taking "How many are A or B" to mean two separate questions combined: "How many are A, how many are B". I can see how that could be tempting in this case; the two numbers happen to be the same, so he could take the question to mean "How many are A which is also the same as the number that are B". If there had been 3 spoons and 4 forks, that interpretation would not have made as much sense; the best answer he could give would be "3, or 4".
We will see how to use the multiplication rule by looking at a few examples. First suppose that we roll a six sided die and then flip a coin. These two events are independent. We see that there are twelve outcomes, all of which are equally likely to occur. The multiplication rule was much more efficient because it did not require us to list our the entire sample space. For the second example, suppose that we draw a card from a standard deck , replace this card, shuffle the deck and then draw again.
We then ask what is the probability that both cards are kings. Since we have drawn with replacement , these events are independent and the multiplication rule applies. The reason for this is that we are replacing the king that we drew from the first time. If we did not replace the king, then we would have a different situation in which the events would not be independent.
The probability of drawing a king on the second card would be influenced by the result of the first card. Actively scan device characteristics for identification. Use precise geolocation data. Select personalised content. Create a personalised content profile.
You have a cowboy hat, a top hat, and an Indonesian hat called a songkok. You also have four shirts: white, black, green, and pink. If you choose one hat and one shirt at random, what is the probability that you choose the songkok and the black shirt? The two events are independent events; the choice of hat has no effect on the choice of shirt. There are three different hats, so the probability of choosing the songkok is 1 3.
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