1 / 18

Precalculus

Precalculus. Counting Principles & Probability. Counting Principles. Fundamental Counting Principle … If E1 can occur in m1 different ways and E2 can occur in m2 different ways, then the # of ways 2 events can occur is m1 x m2.

enoch
Télécharger la présentation

Precalculus

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Precalculus Counting Principles & Probability

  2. Counting Principles • Fundamental Counting Principle … If E1 can occur in m1 different ways and E2 can occur in m2 different ways, then the # of ways 2 events can occur is m1 x m2. • The Fundamental Counting Principle can be extended to 3 or more events. • Example: A diner offers breakfast combination plates which can be made from a choice of one of 4 different types of breakfast meats, one of 8 different styles of eggs, and one of 5 different types of breads. How many different breakfast combination plates are possible?

  3. Counting Principles • Permutation … The ordering of n different elements such that one element is first, one is second, one is third, etc. • # of Permutations of n Elements = n! That is n x (n – 1) x . . . 4 x 3 x 2 x 1 = n! There are n! different ways that n elements can be ordered. • # of Permutations of n Elements taken r at a time ..

  4. Counting Principles • Distinguishable Permutations … Suppose a set of n objects has n1 of one kind, n2 of a second time, and so on with n = n1 + n2 + . . . + nk • # of Combinations of n Elements taken r at a time (Order is NOT important) …

  5. Examples • Determine the # of ways a computer can randomly generate two integers whose sum is 10 …

  6. Examples • A college needs 2 more faculty members: a chemist and an economist. How many ways can these positions be filled if there are 3 applications for the chemistry position and 6 applications for the economics position?

  7. Examples • In Ohio, automobile license plates consist of 2 letters followed by a 4-digit number. How many distinct license plate numbers can be formed?

  8. Examples • There are 5 boys and 4 girls in a group. In how many ways can I select a committee of 4 consisting of 2 boys and 2 girls? • In how many ways can a chairperson, a vice chairperson, and a recording secretary be chosen from a committee of 14 people?

  9. Examples • In how many ways can 5 children line up in a row? • There are 40 #’s in the Ohio lottery. In how many ways can a player select 6 of the #’s?

  10. Examples I • n how many distinguishable ways can the letters COMMITTEE be written? • In how many ways can a research team of 3 students be chosen from a class of 14 students?

  11. Probability • A happening whose result is uncertain is called an experiment. The possible results of the experiment are outcomes, the set of all possible outcomes of the experiment is the sample space of the experiment, and any subcollection of a sample space is an event. • The measure of the likelihood that an event will occur based on chance is called the probability of an event. If event E has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes, the probability of event E is n(E)/n(S) … • The probability of an event must be between 0 and 1.

  12. Probability • Sample Space: The set of all possible outcomes of the experiment. • Independent Events: Occurrence of one event has no effect on the occurrence of the other. • Complement of an Event: The collection of all outcomes that are not in the sample space. • Impossible Event: P(E) = 0 … The event E cannot occur. • Certain Event: P(E) = 1 … The event E must occur.

  13. Probability • Two events A and B (from the same sample space) are mutually exclusive if A and B have no outcomes in common. • If A and B are events in the same sample space, the probability of A or B occurring is given by … • If A and B are mutually exclusive, then …

  14. Probability • If A and B are independent events, the probability that both A and B will occur is … • If A is an event, A’ is its complement, and the probability of A is P(A), then the probability of the complement is …

  15. Examples • Find the probability of getting at least one head if you toss a coin 3 times. • If you toss a 6-sided die twice, find the probability that the sum is odd and no more than 5.

  16. Examples • If you draw 2 marbles (w/o replacement) from a bag containing 1 green, 2 yellow, and 3 red marbles, what is the probability that neither marble is yellow? • Given the probability that an event will happen is 0.7, find the probability that it won’t happen.

  17. Examples • A box contains 3 red marbles, 5 black marbles, and 2 yellow marbles. If 2 marbles are randomly selected with replacement, what is the probability that both marbles are yellow? • A box contains 3 red marbles, 5 black marbles, and 2 yellow marbles. If 2 marbles are randomly selected without replacement, what is the probability that both marbles are yellow?

  18. Examples • A class is given 20 problems from which 10 will be part of an upcoming exam. If a student knows how to solve 15 of the problems, find the probability that the student will be able to answer … A. All 10 questions B. Exactly 8 questions C. At least 9 questions

More Related