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Pre Presentation 5

Pre Presentation 5. Chapter 9 Section 1 and 2 You Try. YOU TRY Problems.

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Pre Presentation 5

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  1. Pre Presentation 5 Chapter 9 Section 1 and 2 You Try

  2. YOU TRY Problems • A tube of sweets contains 10 red sweets, 7 blue sweets, 8 green sweets and 5 orange sweets.If a sweet is taken at random from the tube, what is the probability that it is:(a) red (b) orange (c) green or red (d) not blue

  3. YOU TRY Problems • Nine balls, each marked with a number from 1 to 9, are placed in a bag and one ball is taken out at random. What is the probability that the number on the ball is:(a) odd (b) a multiple of 3 (c) a 5 (d) not a 7

  4. YOU TRY Problems • A card is taken at random from a standard 52-card pack of playing cards.What is the probability that it is: • a) a seven • b) a heart • c) a red card • d) a red six

  5. YOU TRY Problems • You roll a fair dice.What is the probability that the number you get is: • a) a five • b) an odd number • c) a number greater than 1 • d) a multiple of 4?

  6. YOU TRY Problems • Two dice are rolled, find the probability that the sum is a) equal to 1 b) equal to 4 c) less than 13

  7. A die is rolled and a coin is tossed, find the probability that the die shows an odd number and the coin shows a head. • Hint: what is the sample space?

  8. Activity • Students, write down the first letter of your last on a card; now hold them up high for all to see • If one of these cards is taken at random.(a) What is the probability that the letter on it is: A, S, B.. • (b) Which letter is the most likely to be chosen?

  9. Mathematical Induction • You Try • Theorem: • For any positive integer n, 1 + 2 + ... + n = n(n+1)/2. • Proof: • Let's let P(n) be the statement "1 + 2 + ... + n = (n (n+1))/2." (The idea is that P(n) should be an assertion that for any n is verifiably either true or false.) • The proof will now proceed in two steps: the initial step and the inductive step. • Show video

  10. Initial Step. We must verify that P(1) is True. • n = n(n+1)/2 • P(1) asserts "1 = 1(2)/2", which is clearly true. • So we are done with the initial step. • Inductive Step. Here we must prove the following assertion: • "If there is a k such that P(k) is true, then (for this same k) P(k+1) is true." • Thus, we assume there is a k such that 1 + 2 + ... + k = k (k+1)/2. (We call this the inductive assumption.) • We must prove, for this same k, the formula 1 + 2 + ... + k + (k+1) = (k+1)((k+1)+1)/2. So, 1 + 2 + ... + k + (k+1) = k(k+1)/2 + (k+1) = (k(k+1) + 2 (k+1))/2 = ((k+1)(k+2))/2 = [(k+1)((k+1)+1)]/2 The first equality is a consequence of the inductive assumption.

  11. Presentation 5 Chapter 9 – Section 3 Chapter 10 – Section 1

  12. Section 9.3The Bernoullis and Laplace

  13. ChristiaanHuygen’s • Theory of light, double refraction, pendulum clock, falling bodies, mathematical expectation (the value of the chance) • Bernoulli Brothers • James, John, brothers; Nicholas, nephew • ArsConjectandi(The Art of Conjecturing)-4 parts • Book 2- first adequate proof of the binomial theorem for positive integral powers. • Book 4 – first limit theorem of probability theory (Bernoulli’s Theorem – “law of large numbers”)

  14. Bernoulli’s Theorem “law of large numbers” • Do Experiment • Flip a coin 100 times or 200 times or 1000 times. • Observe the number of Heads • Write down your observations. • How many did you get in what number of trials. • What is the probability?

  15. Marquia de L’Hospital • Abraham De Moivre • Pierre Simon Laplace • John Playfair • Mary Fairfax Summerville

  16. Pierre Simon Laplace • Mecanique Celeste, least squares rule, • Memoirs – TheorieAnalytique des Probabilities • The calculus of generating functions (Analysis) • General theory of Probability (theory of probability, limit theorems and mathematical statistics) Probability of an event is the ratio of the number of cases favorable to it , to the number of possible cases P(Event) = number of favorable outcomes total number of outcomes

  17. If A and B are mutually exclusive events (they cannot both occur at the same time) then P(A or B) = P(A) + P(B) • If A and B are two independent events (the occurrence of one does not affect the probability of the other)then P(A and B) = P(A)*P(B) Ex: Regular Deck of cards (52 cards) and Pinochle deck of cards (48 cards) Regular Deck has 4 Aces and Pinochle has 8 Aces Question: What is the probability of getting an Ace from a Regular and Pinochle deck P(A and B) = (4/52)*(8/48) = (1/78)

  18. Bernoulli process / Bernoulli trials experiment. • The probability of success on a single trial is denoted p and q = 1-p is the probability of failure • P and q remain constant from trial to trial. • Bernoulli showed that the probability of observing exactly r successes in n such trials was expressed by the rth term of the expansion for (p+q)n n (p)r(q)n-r r

  19. Example • Toss 6 dice • What is the probability that exactly 2 of the dice have top faces showing at least a five? • the probability of success is 1/3 = p • (a five or a six showing) • The probability of failure q = 2/3 = 1-p = 1-(1/3) • P(2 successes and 4 failures) = 6 (1/3)2 (2/3q)4 2 15(1/9)(16/81) = 240/729

  20. Daniel Bernoulli, Proceedings of the St. Petersburg Academy of Science • Simeon Denis Poisson • PafnutyChebyshev, central limit theorem • See video • Andrei Markov

  21. Chapter 10The Revival of Number Theory: Fermat, Euler and Gauss Section 10.1Marin Mersenne and the Search for Perfect Numbers

  22. Marin Mersenne • Finding all perfect numbers • Definition: A positive integer n is said to be perfect if n is equal to the sum of all its positive divisors not including n itself • Example: 6 is a perfect number • 6 {1,2,3,6} hence, 6=1+2+3 We could say (n) = 2n; example (6) = 2*6 = 1+2+3+6 =2*6 You try, find another perfect number. 28

  23. Euclid’s Elements • 1+2+22 + 23+…+ 2(k-1) = p Where p is a prime number, then 2(k-1) p is a perfect number Ex: 1+2+4= 7 is a prime, hence, 4*7 = 28 is a perfect number Theorem: If 2(k-1) (2k -1) is prime (k>1) then n = 2(k-1) (2k -1) is a perfect number. Further more, (n) = 2(k) (2k -1)= 2*2(k-1) (2k -1) = 2n the effect of which is to make n a perfect number.

  24. Mersenne Prime Number • A Mersenne Number is a number that is equal to one less than a power of 2, or (2n-1) where n is any positive integer. • A Mersenne Prime Number is a prime number that happens to also be a Mersenne Number. • A Mersenne Prime Number will always have a value equal to (2n-1) where n is one of a selected list of positive prime numbers. • Example: 22-1 = 323-1 = 725-1 = 3127-1 = 127

  25. Amicable numbers • Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. • A proper divisor of a number is a positive integer divisor other than the number itself. • For example, the proper divisors of 6 are 1, 2, and 3. • For example, the smallest pair of amicable numbers is (220, 284); • The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; • The proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220. • Can you find another amicable pair?

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