Probability: The Study of Randomness Randomness and Probability Models. IPS Chapters 4 Sections

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1 Probability: The Study of Randomness Randomness and Probability Models IPS Chapters 4 Sections

2 Chapter 4 Overview Key Concepts Random Experiment/Process Sample Space Events Probability Models Probability of Events Laws of Probability Law of the Compliment Addition Law (Disjoint Events) Multiplication Law (Independent Events) Random Variables Discrete Continuous Probability Distributions Mean & Standard Deviation of Discrete Random Variables Mean & Standard Deviation of Continuous Random Variables Skills Learned Constructing Probability Models Computing Probability of Events Applying the Laws of Probability Constructing the Probability Distribution of Discrete Random Variables Computing the Mean & Standard Deviation of Discrete Random Variables Computing Probabilities for Discrete Random Variables Computing Probabilities for Continuous Random Variables

3 Randomness and probability Random Experiments/Processes/Phenomena A phenomenon is random if individual outcomes are uncertain (unpredictable), but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

4 The Craps Table

5 The Pass line bet Pass line bet: The fundamental bet in craps is the pass line bet, which is a bet for the shooter to win their point number. A pass line bet wins outright if the come-out roll is a 7 or 11. If the come-out roll is 2, 3 or 12, the bet loses immediately (known as "crapping out"). If the roll is any other value, the shooter establishes a point {e.g., 4, 5, 6, 8, 9, 10}; if the point is rolled again before a seven, the bet wins. If, with a point established, a seven is rolled before the point is rerolled, the bet loses ("seven out"). A pass line win pays even money (1:1). Question: What is the probability that the player wins the Pass line bet outright?

6 Empirical Concept of Probability 1. Repeat an experiment (or observe a random phenomenon) a large number of times. 2. Record the number of times a desirable outcome occurs (e.g., the player rolls a 7.) 3. Compute the ratio: # of times the event ocurred Total# of times the experiment was performed Empirical Concept of Probability The probability of any outcome of a random phenomenon can be defined as the proportion of times the outcome would occur in a very long series of repetitions.

7 Simulating the toss of a single die TI-83/84 randint(1,6) Generates a random integer between 1 & 6 (inclusive) randint(1,6,10) Generates 10 random integers between 1 & 6 (inclusive) randint(1,6,10) L 1 Generates 10 random integers between 1 & 6 (inclusive) and stores them in L 1

8 Example: Coin toss The result of any single coin toss is random and unpredictable. But the relative frequency of each outcome over a long run (many tosses) is predictable. The probability of heads is 0.5 = the proportion of times you get heads in many repeated trials. First series of tosses Second series

9 Probability Models 1. Random experiment 2. Sample Space All possible outcomes S={e 1, e 2, e 3,, e n } Example: Probability Model for a Coin Toss S = {Head, Tail} Outcome Heads Tails Probability Probability Distribution Function Assignment of probabilities to the outcomes in the sample space Outcome e 1 e 2 e 3... e n Probability p 1 p 2 p 3... p n

10 Probability Rules Rule 1 The probability of each outcome is a number between 0 and 1 (inclusive). 0 p{e k } 1 Rule 2 The sum of the probabilities of all the outcomes in a sample space equals 1. p(e k ) = 1

11 Example: Tossing of a Die Sample Space S = { } Probability Distribution Function Outcome Probability 1/6 1/6 1/6 1/6 1/6 1/6

12 Example Tossing a Pair of Dice Random Experiment: A pair of dice is tossed. The Sample Space: All possible outcomes of the experiment. S = { } In the game of craps, since a player is only interested in the sum of the two numbers that show up (and not on what number shows up on the face of each die), we can express the sample space as: S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

13 Example Tossing a Pair of Dice Probability Distribution Function: Assignment of probabilities to all the outcomes of the experiment.

14 Events Most often we are not just interested in a single outcome of an experiment, we are interested in a collection of outcomes. For example, in the game of craps, a player loses outright if s/he rolls a 2, 3, or 12. The player wins if s/he rolls a 7 or 11. An event is a subset of the Sample space (a sub collection of all the possible outcomes). E = The player loses a bet outright = {2, 3, 12} F = The player wins a bet outright = {7, 11}

15 Probability of Events Rule 3 The probability of an event is the sum of probabilities of all the outcomes that make up the event. Examples: What is the probability of the following events? E = The player loses a bet outright = {2, 3, 12} F = The player wins a bet outright = {7, 11} Answers P(E) = p(2) + p(3) + p(12) = 1/36 + 2/36 + 1/36 = 4/36 or 1/9 = or 11.11% P(F) = p(7) + p(11) = 6/36 + 2/36 = 8/36 or 2/9 = or 22.22%

16 Disjoint Events In the game of craps, the events E = The player loses a bet outright and F = The player wins a bet outright have no outcomes in common and cannot both occur when a player rolls a pair of dice. E = The player loses a bet outright = {2, 3, 12} F = The player wins a bet outright = {7, 11}

17 Addition Rule for Disjoint events Rule 4 If two events are disjoint, then the probability that one or the other will occur is the sum of the individual probabilities of the events. P(A or B) = P(A U B) = P(A) + P(B) Example: What is the probability that in the game of craps the player will neither win or lose outright? E = The player loses a bet outright = {2, 3, 12} P(E) = 1/9 F = The player wins a bet outright = {7, 11} P(F) = 2/9 P(E or F) = 1/9 + 2/9 = 3/9 or 1/3 =.3333 or 33.33%

18 Law of the Complement Rule 5 The complement of any event A is the event that consists of all the outcomes not in A, written as A c (or not A) The complement rule states that the probability of the complement of an event is 1 minus the probability of the event. P(not A) or P(A c ) = 1 P(A) Venn diagram: Sample space made up of an event A and its complementary A c, i.e., everything that is not A. Example In the game of craps, the probability the player will roll a 7 is 6/36 or 1/6. The probability that the player will not roll a 7 is 1 1/6 or 5/6.

19 Multiplication Rule for Independent Events Rule 6 Two events A and B are independent if knowing that one occured does not change the probability that the other occurs. If A and B are independent, P(A and B) = P(A)P(B) Example: In the game of craps, the probability that the player will roll double sixes equals the probability that a six will show up on he first die and on the second die. P(6 and 6) = P(6 on first die) * P(6 on second die) = 1/6 * 1/6 = 1/36

20 Examples A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors. 1. What is the probability that the ball will land in any one slot? 2. If you bet on red, you win if the ball lands in a red slot. What is the probability of winning? 3. The slot numbers are laid out on a board on which gamblers place their bets. One column of numbers on the board contains all multiples of 3, that is, 3, 6, 9,, 36. You place a column bet that wins if any of these numbers comes up. What is your probability of winning? Answers: (1) 1/38 (2) 18/38 (3) 12/38

21 Q1: A couple wants to have three children. What are the possible arrangements of boys (B) and girls (G) the couple can have? Assuming that the probability that a baby is a boy or a girl is the same, 0.5, what is the probability of each outcome. Sample space: {BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG} All eight outcomes in the sample space are equally likely. The probability of each is thus 1/8. OR Each birth is independent of the next, so we can use the multiplication rule. Example: P(BBB) = P(B)* P(B)* P(B) = (1/2)*(1/2)*(1/2) = 1/8 Q2: A couple wants three children. What are the possible numbers of girls (X) they could have? What is the probability of each outcome? The same genetic laws apply. We can use the probabilities above and the addition rule for disjoint events to calculate the probabilities for X. Sample space: {0, 1, 2, 3} P(X = 0) = P(BBB) = 1/8 P(X = 1) = P(BBG or BGB or GBB) = P(BBG) + P(BGB) + P(GBB) = 3/8 Continue in that way to compute P(X=2) & P(X=3)

22 Dice You toss two dice. What is the probability of the outcomes summing to 5? This is S: {(1,1), (1,2), (1,3), etc.} There are 36 possible outcomes in S, all equally likely (given fair dice). Thus, the probability of any one of them is 1/36. P(the roll of two dice sums to 5) = P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 / 36 = 0.111

23 Simulating the toss of a single die with Excel Copy the formula to a 10 x 10 grid to simulate 100 tosses of the die.

24

25 Simulating the Toss of a pair of Dice Copy the formula to a 20 x 10 grid to simulate 200 tosses of a pair of dice.

26 Simulation of 400 tosses of a pair of dice

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