Theoretical Probability of Simple Events

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Theoretical probability is the probability that an event occurs when all of the outcomes of the experiment are equally likely. The probability of an event is the ratio of the number of ways an event can occur to the total number of equally likely outcomes. A probability model provides a probability for each possible non-overlapping outcome for a chance process so that the total probability over all such outcomes is 1. The collection of all possible outcomes is called the sample space, and if the probabilities are based on the structure of the process and its outcomes, then the probabilities are theoretical.

To find the theoretical probability of a simple event write the ratio of the number of ways the event can occur to the total number of possible outcomes. Both theoretical and experimental probability are between 0 and 1. The theoretical probability of an event is fixed while the experimental probability will change based on the experiment. The formula for the complement is the same for both probability definitions. Theoretical probability is based on the structure of an experiment while experimental probability is based on the results of the experiment.

The Law of Large Numbers is an interesting theorem in probability theory. The theorem relates experimental probability and theoretical probability. The basic idea of the theorem is that as an experiment is repeated many times, the experimental probability of the event approaches the theoretical probability of the event. For example, consider flipping a coin ten times. The theoretical probability of heads is 1 2 . The individual flipping of the coin ten times may not get five heads and five tails. However, after flipping the coin many more times, perhaps 100 times, the experimental probability will tend to get closer to 1 2 .

You and your friend sell T-shirts You and your friend sell T-shirts. The table shows the types you sell and how many of each you have. Explain how to find the probability of choosing a blue sports logo T-shirt at random.

A bag contains 4 white tiles and 8 black tiles A bag contains 4 white tiles and 8 black tiles. You select one tile at random from the bag. What is the probability that you select a black tile? Write your answer in simplest form.

Find the probability the spinner lands on blue or green.

A bowl of flower seeds contains 5 petunia seeds and 15 begonia seeds A bowl of flower seeds contains 5 petunia seeds and 15 begonia seeds. Riley calculated the probability that a randomly selected seed is a petunia seed as 1 3 . Describe and correct Riley’s error.

There are 20 seventh graders and 15 eighth graders in a club There are 20 seventh graders and 15 eighth graders in a club. A club president will be chosen at random. Compare the probabilities of choosing a seventh grader or an eighth grader.

Exit Ticket Each card in a set of cards has one of the letters from the word mathematics. The cards are shuffled. What is the probability of drawing the letter m? What is the probability of drawing a vowel? What is the probability of drawing a consonant? Suppose you draw a card from the set of cards, record the letter, return the card to the set, and shuffle the cards. You repeat this experiment 22 times. Would you expect the experimental probability of drawing a vowel to be the same as the theoretical probability? Explain.