PSYC 301 1st Edition Lecture 3Measuring ProbabilityOutline of Last Lecture II. Characterize the center of a distribution of dataIII. Calculate how much scores varyOutline of Current Lecture I. Measure probability II. Learn why random sampling is a good thingIII. Calculate probability from a histogramCurrent LectureI. Probability – “What are the odds of X happening?”a. Expected relative frequency of a particular outcomei. Outcome: one of the events that could happenProbability =Possible Successful OutcomesAll Possible OutcomesII. Probability vs. PercentageProbabilities PercentagesP x %0 ≤ p ≤ 1 0% ≤ x ≤ 100%Never Negative! Never Negative!Sum to 1 Sum to 100%III. Measuring Probability i. List or count all possible outcomesii. How many different ways are there of achieving an outcome?These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.1. Sometimes there is only one way, other times there are manyiii. What is the outcome of interest?b. Example: Flipping a coin ONE timei. A coin has 2 sides, heads or tails, either oneof those is your OUTCOME. ThePROBABILITY of you landing on heads is ½.The PROBABILITY of you landing on tails is ½,also. c. Example: Flipping a coin TWO timesi. You have 4 possible OUTCOMES: If the firsttime you toss the coin and it lands on heads,the second time it could land on eitherheads or tails. Or is the first time you tossthe coin and it lands on heads, the secondtime it could land on heads or tails. Theprobability of each of those 4 outcomes tohappen is 25%. d. Example: Rolling a die ONE time- You can roll a 1, 2, 3, 4, 5, or 6- You have a 1/6 chance to roll a 1, a 1/6 chance to roll a 2, a 1/6 chance to roll a 3, etc…e.f.g.h.i.j.k.l.e. Example: RollingTWO die at the same timea. If one dice rolls a 1, the second die can be a1, 2, 3, 4, 5, or 6i. If one dice rolls a 2, the second diecan be a 1, 2, 3, 4, 5, or 6ii. If one dice rolls a 3, the second diecan be a 1, 2, 3, 4, 5, or 6… etc.b. The outcome equals the dice addedtogether. The probability will vary becausethere is a different number of ways each outcome is reached.Outcome ProbabilityHeads 0.5 (1/2)Tails 0.5 (1/2)Outcome ProbabilityHH 0.25 (1/4)HT 0.25 (1/4)TT 0.25 (1/4)TH 0.25 (1/4)Outcome Probability1 0.167 (1/6)2 0.167 (1/6)3 0.167 (1/6)4 0.167 (1/6)5 0.167 (1/6)61.167 (1/6)Outcome Probability2 0.03 (1/36)3 0.06 (2/36)4 0.08 (3/36)5 0.11 (4/36)6 0.14 (5/36)7 0.17 (6/36)8 0.14 (5/36)9 0.11 (4/36)10 0.08 (3/36)11 0.06 (2/36)12 0.03 (1/36)i. For example: you have to roll a 1 and a 1 to reach an outcome of 2. There are no other ways to reach an outcome of 2. To have an outcome of 7, you can roll a 1 & 6, you can roll a 2 & 5, or you can roll a 3 & 4.f. Measuring the Probability of Picking a random card out of a deck.a. There are 52 cards in a full deck. There are 4 suits (C, S, H, D), and there are 13 different cards. Meaning you have 4 suits per card.i. If your outcome is any card, the Probability of you pulling any random card is the same: 0.08 (4/52). You have 4 of each card / 52 cards in the deckIV. Random Sampling- Law of Large Numbersa. This does not mean that the outcome is unpredictableb. Random processes usually produce highly regular resultsc. Nonrandom processes won’t be representative of the populationd. The Probability of a random outcome can be estimated by the proportion of times it happens in a very large number of trialsV. Calculating Probability (3 ways)1. Independent Trials: when the outcome of one trial does not affect the outcome of the next.- The Probability of your outcome will never change1) Examples: tossing a coin, rolling a die, picking cards from a stack (as long as the card is replaced before drawing another) - If two events are independent, then the probability of A and B (in that order) is p(A)*p(B)1) Two trials in succession:i. Flipping a Coin: A=heads, B=tails → ½ x ½ = ¼ii. Rolling one die: A=1, B=3 → 1/6 x 1/6 = 1/36iii. Rolling two dice: A=6, B=12 → 5/36 x 1/36 = 5/1296iv. Picking a card: A=8, B=Q → 4/52 x 4/52 = 16/2704- If two events A and B have no outcomes in common (are disjointed/ mutually exclusive), the probability of A or B is p(A) + p(B)1) Only one trialOutcome ProbabilityHeart 0.25 (13/52)Diamond 0.25 (13/52)Club 0.25 (13/52)Spade 0.25 (13/52)ii. Flipping a Coin: A=heads, B=tails → ½ + ½ = 1iii. Rolling one die: A=1, B=3 → 1/6 + 1/6 = 2/6iv. Rolling two dice: A=6, B=12 → 5/36 + 1/36 = 6/36v. Picking a card: A=8, B=Q → 4/52 + 4/52 = 8/52- The probability that an event does NOT occur is 1 minus the probability of that event [ 1 = (all possible outcomes) – (# or % of it happening) ]1) One triali. Flipping a Coin: NOT flipping heads → 1 – ½ =1/2ii. Rolling one die: NOT rolling a 2 → 1 – 1/6 = 5/6iii. Rolling two dice: Not rolling a 7 → 1 – 6/36 = 30/36iv. Picking a card: NOT picking a spade → 1 – 13/52 = 39/522. Histograms show all possible outcomes. All outcomes have a probability that sumis equal to 1 / The sum of all probabilities in a histogram equals 1.- We can use a histogram to find out the probability of some particular events happening.‘ ‘ ‘ ‘ ‘ ‘Probability1/6 - -0 1 2 3 4 5 6Die Roll- What is the probability of rolling something greater than 4?- Any number greater than 4 is either a 5 or a 6. There is a 1/6 chance you will roll a 5 and a 1/6 chance you will roll a 6. [1/6+1/6=2/6]- What is the probability of rolling something less than 4?- 1, 2, and 3, are all less than 4. You have 1/6th of a chance to roll a 1, a 2, or a 3. [1/6+1/6+1/6=3/6]- What is the probability of rolling a number larger than 2 and less than 6?- Larger than 2 and less than 6 leaves you with 3, 4, and 5. You have 1/6th of a chance of hitting either 3 numbers. [1/6+1/6+1/6=3/6]0123456Mean- Using a histogram like this one is easy to calculate percent’s. - What is the probability of selecting a value between 0 and 1?- The space between the 0 and the 1 on the x-axis is 34%- What is the probability of selecting a value above 1?- 14% + 2% = 16%VI. Questionsa. What is the full range of values that can equal the probability? i. 0 → 1b. How do you calculate the probability of two independent events occurring in a particular order?i. Multiply the two togetherc. In a
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