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VOCAB REVIEW: Pascal’s triangle - A triangular arrangement of where each row corresponds to a value of n. Pascal’s Triangle n = 0; 20 This is a Pascal’s Triangle – Each row is labeled as n • The first row is n = 0 • Second is n = 1 etc… – Each term is nCr • n is the row • r is the position in the row starting with 0 – Each term is the sum of the two directly above it. n = 1; 21 n = 2; 22 n = 3; 23 n = 4; 24 1. The 6 members of a Model UN student club must choose 2 representatives to attend a state convention. Use Pascal’s triangle to find the number of combinations of 2 members that can be chosen as representatives. 2. Use Pascal’s triangle again to find the number of combinations of 2 members that can be chosen if the Model UN club has 7 members. Binomial Theorem The Binomial Theorem • Gives us the coefficients for a binomial expansion • The values in a row of Pascal's triangle are the coefficients in a binomial expansion of the same degree as the row. • A binomial expansion of degree n is (a + b)n. • The variables are anb0 + an-1b1 + … + a1bn-1 + anb0 + a0bn Expand a Power of a Binomial Sum x 1st 2 y term 2nd term Pascal’s # 3 x x xx 1x 26 3 24 2 1 22 2 0 y1 y yy yy 0 1 1 3 22 33 3 1 x 3x y 3x y y 6 4 2 2 3 a 2b 1st term 2nd term Pascal’s # 4 a aa a aa 1a 0 1 2 2 3 3 44 2b 21b -22bb 42bb -82bb 16 44 1 33 4 22 6 1 0 4 1 a 8a b 24a b 32ab 16b 4 3 2 2 3 4 Use the binomial theorem to write the binomial expansion. 1. x 3 2. 2 p q 5 4 3. a 2b 4 4. 5 2 y 3 Find a Coefficient in an Expansion • Find the coefficient of x in the expansion of p q n ax by m where rm/p r C (1 st ) (2 nd ) n r nr • Find the coefficient of x⁴ in the expansion of (3x + 2)¹º. n= r= x5 Binomial Formula 1. Use the binomial formula to find the coefficient of the 10 in the expansion of q 3z q9 z 2. Find the coefficient of the x5 in the expansion of (x – 3)7? 3. Find the coefficient of the x3 in the expansion of (2x +5)8? term 12.3 An Introduction to Probability What do you know about probability? • Probability is a number from 0 to 1 that tells you how likely something is to happen. • Probability can have two main approaches -experimental probability -theoretical probability Experimental vs.Theoretical Experimental probability: P(event) = number of times event occurs total number of trials Theoretical probability: P(E) = number of favorable outcomes total number of possible outcomes How can you tell which is experimental and which is theoretical probability? Experimental: You tossed a coin 10 times and recorded a head 3 times, a tail 7 times P(head)= 3/10 P(tail) = 7/10 Theoretical: Toss a coin and getting a head or a tail is 1/2. P(head) = 1/2 P(tail) = 1/2 Experimental probability Experimental probability is found by repeating an experiment and observing the outcomes. P(head)= 3/10 A head shows up 3 times out of 10 trials, P(tail) = 7/10 A tail shows up 7 times out of 10 trials Theoretical probability HEADS TAILS P(head) = 1/2 P(tail) = 1/2 Since there are only two outcomes, you have 50/50 chance to get a head or a tail. How come I never get a theoretical value in both experiments? Tom asked. • If you repeat the experiment many times, the results will getting closer to the theoretical value. • Law of the Large Numbers Experimental VS. Theoretical 54 53.4 53 52 51 50 49 50 49.87 48.4 48 47 46 45 1 48.9 Thoeretical 5-trial 10-trial 20-trial 30-trial Law of the Large Numbers 101 • The Law of Large Numbers was first published in 1713 by Jocob Bernoulli. • It is a fundamental concept for probability and statistic. • This Law states that as the number of trials increase, the experimental probability will get closer and closer to the theoretical probability. http://en.wikipedia.org/wiki/Law_of_large_numbers Contrast experimental and theoretical probability Experimental probability is the result of an experiment. Theoretical probability is what is expected to happen. You must show the probability set up, the unreduced fraction, and the reduced fraction in order to receive full credit. Geometric Probability • Geometric probabilities are found by calculating a ratio of two side lengths, areas, or volumes according to the problem. Find a Geometric Probability • You throw a dart at the square board. Your dart is equally likely to hit any point inside the board. Are you more likely to get 10 points or 0? (use area) 2 5 10 0 • HW 37: pg 719, 13-43 odd