Aim: Who is Bernoulli and what is his experiment regarding probability?

1. Aim: Who is Bernoulli and what is his experiment regarding probability? Do Now: Find the probability of obtaining exactly 2 odd numbers on successive spins.

2. Probability of Two Outcomes Find the probability of obtaining exactly 3 odd numbers on successive spins. 4 odd numbers on successive spins.

3. Probability of Two Outcomes Find the probability of obtaining exactly 1 odd number on 4 spins of the arrow 4 possible ways to spin exactly 1 odd number

4. combination of 2 odd numbers out of 4 spins Probability of Two Outcomes Find the probability of obtaining exactly 2 odd number on 4 spins of the arrow 6 possible ways of spinning exactly 2 odd

5. combination of 3 odd numbers out of 4 spins Probability of Two Outcomes Find the probability of obtaining exactly 3 odd number on 4 spins of the arrow

6. 4 possible ways to spin exactly 1 odd number 6 possible ways to spin exactly 2 odd numbers 4 possible ways to spin exactly 3 odd numbers Probability of Two Outcomes Find the probability of obtaining exactly ? odd number on 4 spins of the arrow What are the two outcomes here? odd and not odd probability experiments w/exactly 2 possible outcomes - binomial probabilityor a Bernoulli experiment

7. Probability of Success & Failure Consider n independent trials of an experiment where each trial has two possible outcomes called success and failure. The probability of success on each trial is p and the probability of failure is . nCk pkqn – k gives the probability of k successes in the n trials of the experiment q = 1 –p A fair coin is tossed seven times. Find the probability of obtaining four heads.

8. Probability of Success & Failure Consider n independent trials of an experiment where each trial has two possible outcomes called success and failure. The probability of success on each trial is p and the probability of failure is q = 1 –p. nCk pkqn – k gives the probability of k successes in the n trials of the experiment A fair coin is tossed seven times. Find the probability of obtaining four heads. What is q? What is p? 1/2 q = 1 –p = 1/2 What is n? 7 What is k? 4

9. Model Problem If a fair coin is tossed 10 times, what is the probability that it falls tails exactly 6 times? nCk pkqn– k probability of success p = P(tails) = ½ probability of failure q = P(not tails) = ½ n = 10 trials r = 6 number of successes

10. Model Problem If 5 fair dice are tossed, what is the probability that they show exactly 3 fours? nCk pkqn– k probability of success p = P(4) = 1/6 probability of failure q = P(not 4) = 5/6 n = 5 trials r = 3 number of successes

11. Aim: Who is Bernoulli and what is his experiment regarding probability? Do Now: If 6 fair dice are tossed, what is the probability that they show exactly 4 threes?

12. Probabilities must be added At least David is rolling 5 dice. To win at this game, at least three of the 5 dice must be “ones”. David, therefore, can win by rolling 3, 4 or 5 “ones”. What is the probability David can win? In General: At least r successes in n trials means r, r + 1, r + 2, . . . , n successes.

13. At Least Rose is the last person to compete in a basketball free-throw contest. To win, Rose must be successful in at least 4 out of 5 throws. If the probability that Rose will be successful on any single throw is 3/4, what is the probability that Rose will win the contest? Rose must be successful in 4 or in 5 throws.

14. At Least Rose is the last person to compete in a basketball free-throw contest. To win, Rose must be successful in at least 4 out of 5 throws. If the probability that Rose will be successful on any single throw is 3/4, what is the probability that Rose will win the contest? P(at least 4 out of 5 successes) = P(4 of 5) + P(5 of 5)

15. At Most A family of 5 children is chosen at random. What is the probability that there are at most 2 boys in this family of 5 children? Success means to have 0, 1, or 2 boys. no boys: 1 boy: 2 boys:

16. At Most A family of 5 children is chosen at random. What is the probability that there are at most 2 boys in this family of 5 children? Success means to have 0, 1, or 2 boys. P(at most 2 boys out of 5 children) = P(0 of 5) + P(1 of 5) + P(2 of 5)

17. At least • A coin is loaded so that the probability of heads is 4 times the probability of tails. • What is the probability of heads on a single throw? • What is the probability of at least 1 tail in 5 throws? 4x + x = 5x P(heads) = 4x/5x = 4/5 P(tails) = x/5x = 1/5

18. At least A coin is loaded so that the probability of heads is 4 times the probability of tails. b. What is the probability of at least 1 tail in 5 throws? P(at least 1 tail in 5 throws) = P(1 t) + P(2 t) + P(3 t) + P(4 t) + P(5 t)

19. Alternate Solution A coin is loaded so that the probability of heads is 4 times the probability of tails. b. What is the probability of at least 1 tail in 5 throws? success or what is Probability of all heads? failure P(success) = 1 – P(failure) P(allheads)

20. 4 pt. Regents Question The probability that the Stormville Sluggers will win a baseball game is 2/3. Determine the probability, to the nearest thousandth, that the Stormville Sluggers will win at least 6 of their next 8 games.

21. 4 pt. Regents Question A study shows that 35% of the fish caught in a local lake had high levels of mercury. Suppose that 10 fish were caught from this lake. Find, to the nearest tenth of a percent, the probability that at least 8 of the 10 fish caught did not contain high levels of mercury.