Section 6.1.1 Simulation AP Statistics

AP Statistics, Section 6.1, Part 1 Independence In order for an event to be considered random, it must be independent. That is, it must not be influenced by other (perhaps previous) events. Example: Flipping a head does not make it more probable that a tail will occur next. AP Statistics, Section 6.1, Part 1

AP Statistics, Section 6.1, Part 1 Simulations – Example 6.3 (pg 394) Simulation is a way to model random events, such that simulated outcomes closely match real-world outcomes. Step 1: State the problem Toss a coin 10 times What is the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails? Step 2: State the assumptions A head or tail is equally likely to occur on each toss Tosses are independent of each other AP Statistics, Section 6.1, Part 1

Simulations – Example 6.3 (pg 394) Step 3: Assign digits to represent outcomes. (in the random digit table, digits are either odd or even, 50% chance) One digit simulates one toss of the coin Odd digits represent heads, even digits represent tails Successive digits in the table simulate successive independent tosses of the coin AP Statistics, Section 6.1, Part 1

Simulations – Example 6.3 (pg 394) Step 4: Simulate many repetitions Looking at 10 consecutive digits in the table simulates one repetition of our experiment Starting with line 101: 1 9 2 2 3 9 5 0 3 4 H H T T H H H T H T Were there at least 3 consecutive heads or 3 consecutive tails? yes AP Statistics, Section 6.1, Part 1

Simulations – Example 6.3 (pg 394) Step 4: Simulate many repetitions look at the next 10 digits 0 5 7 5 6 2 8 7 1 3 T H H H T T T H H H Were there at least 3 consecutive heads or 3 consecutive tails? yes AP Statistics, Section 6.1, Part 1

Simulations – Example 6.3 (pg 394) Step 4: Simulate many repetitions look at the next 10 digits 9 6 4 0 9 1 2 5 3 1 H T T T H H T H H H Were there at least 3 consecutive heads or 3 consecutive tails? yes AP Statistics, Section 6.1, Part 1

Simulations – Example 6.3 (pg 394) Step 4: Simulate many repetitions look at 22 more repetitions and record your results How many times did you have at least 3 consecutive heads or 3 consecutive tails? 23 AP Statistics, Section 6.1, Part 1

Simulations – Example 6.3 (pg 394) Step 5: State your conclusions we estimate the probability of a run of size 3 by the proportion Estimated probability = 23/25 = 0.92 Note that we only used 25 trials which is a relatively small number, so our confidence that the true probability is 0.92 is weak. AP Statistics, Section 6.1, Part 1

Create the random numbers MATH PRB 5: (LOWEST DIGIT, HIGHEST DIGIT, HOW MANY) STO> LIST AP Statistics, Section 6.1, Part 2

Simulations – assigning digits Example 6.4 ( pg 395) Example 6.5 (pg 396) Example 6.6 (pg 397) How often does a number of 2 or less appear? AP Statistics, Section 6.1, Part 1

Permutation vs. Combination Permutation - Permutation: A set of objects in which position (or order) is important. (order matters) nPr Combination - Combination: A set of objects in which position (or order) is NOT important. nCr AP Statistics, Section 6.1, Part 1

Permutation vs. Combination Example - What is the total number of possible 4-letter arrangements of the letters   m, a, t, h,  if each letter is used only once in each arrangement? 4nPr4 – order DOES matter since you cannot have mmmm or aata = 24 AP Statistics, Section 6.1, Part 1

Permutation vs. Combination Example - There are 12 boys and 14 girls in Mrs. Schultzkie's math class.  Find the number of ways Mrs. Schultzkie can select a team of  3 students from the class to work on a group project.  The team is to consist of 1 girl and 2 boy 12nCr1 and 12nCr 2 – order DOES NOT matter = 924 AP Statistics, Section 6.1, Part 1

AP Statistics, Section 6.1, Part 1 Assignment Exercises: 6.1 - 6.6 AP Statistics, Section 6.1, Part 1