Probability
Probability: is a numerical measure of the likelihood that an event will occur An experiment: is any process that generates well-defined outcomes Sample space (S): is the set of all possible outcomes of an experiment An event (A): is an outcome or set of outcomes that are of interest to the experiment. An event (A) is a subset of the sample space (S) The probability of an event A {P (A)}: is a measure of the likelihood that an event A will occurExample: Tossing a coin Experiment: Toss a coin and observe the up face S { } S= {H, T} H (head) T (tail) Example: Tossing a coin twice Experiment: flip a coin twice and observe the sequence (keeping track of order) of up faces. S= {HH, HT, TH, TT} A= {Tossing at least one head} A = {HH, HT, TH}
Example = Tossing by a dice Experiment: Tossing a six-sided dice and S= {1, 2, 3, 4, 5, 6} A= {roll an even number} A = {2, 4, 6}
Methods of assigning probability
Classical probability: Each outcome is equally likely It is applicable to games of chance In the cases, if there are N outcomes in S, then the probability of any one outcome is 1/N If A is any event and nA is the number of outcomes in A, then: P (A) =Example: Tossing a dice: S= {1, 2, 3, 4, 5, 6} P (1) = P(2)= P(3)=P (4)=P(5)=P(6)= A= {roll an even number}= {2, 4, 6} P (A) = 3/6 = 0.5
Empirical probability is simply the relative frequency that some event is observed to happen (or fail). Number of times an event occurred divided by the number of trials: P (A) = Where: N= total number of trails nA Number of outcomes producing A
Relative frequency example
Children No.Frequency
Relative frequency
0
40
40/215 = 0.19
1
80
80/215 = 0.37
2
50
50/215 = 0.23
3
30
30/215 = 0.14
4
10
10/215 = 0.05
5
5
5/215 = 0.02
Sum
215
215/215 = 1.00
Basic concepts of probability:
Probability values are always assigned on a scale from 0 to 1 A probability near 0 indicates an event is unlikely to occur A probability near 1 indicates an event is almost certain to occur A probability near of 0.5 indicates event is just as likely as it is unlikely The sum of the probabilities of all outcomes must be 1Definitions
Mutually exclusive events: occurrence of one event precludes the occurrence of the other event Independent event: occurrence of one event does not affect the occurrence or non- occurrence of the other event Complementary events: all elementary events that are not in the event A are in its complementary event. P (Sample space) P (A') = 1-P (A)Laws of Probability
The addition rule: The probability of one event or another P (A or B) = P (A) + P (B) – P (A and B)If A and B are mutually exclusive events (A and B can not occur at the same time), thenP (A or B) = P (A) + P (B)Examples:
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@Ȁ挀⓰缀Ѐ耀저篧뼖뼀āᔀࠀ뼀Ȁ牟汥⽳爮汥汳쇏썪ర惻惯彴왐펈ꅛ틗耾閱Ⳅ貶뉤穧읪銎㿸썉⟡暭턣�껂쇫戜⽜럇』嵚渎䳥渖炤弘號괳뚮⺐ꢱ䩩ୖ굋淌攦⨨ᠲ깆ᬥ붻۳䘹䩟뇭炋죓鉉ꕌ�藆嬊坚᱔�셆穫뺝꾘쉼睯瓰ꑓ�劳쎗畾舋翩㏊凪蠫ﬦ厇毮솽韏ᩨ䄭縮Ͽ倀ŋⴂ᐀ࠀ℀�蔀ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀묀��豈ༀ܀搀獲搯睯牮癥砮汭䭐·̌ඍශwྟྨ〱0ꄀ㈏Ѐਈ܀̀最ȀȀȀᰀǾ䌄ȄȀᰀꨀਏЀĀꘀఏ퐀퀁ဃ༅Ѐ런ሀࣰ℀@Ȁ挀⓰缀Ѐ耀䠀篣뼖뼀āᔀࠀ뼀Ȁ牟汥⽳爮汥汳쇏썪ర惻惯彴왐펈ꅛ틗耾閱Ⳅ貶뉤穧읪銎㿸썉⟡暭턣�껂쇫戜⽜럇』嵚渎䳥渖炤弘號괳뚮⺐ꢱ䩩ୖ굋넊仱̠깁뫲땱罹౦㈢贺瞭ᅄ诋ᘙῚ諝欎䖮뢂ꁘ쌂ᗜ왒郊㣅ᷰ풹糭좰蘩잀뜄Ჭ�婈屬ず퇘ꆃ뽪뇾㸊믲�靾﹃⩼﮷䦷辕䯆꺥絛⧻ꘈﾞ禟㲯庣芵뉤㥻ꆕ⯑䲌糡ꙉዉﳤϿ倀ŋⴂ᐀ࠀ℀�蔀ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀�鸿홺豈ༀ܀搀獲搯睯牮癥砮汭䭐·̊ඍჀශvྟྨ㔵ྡ2ࠀg︀ЀCЀྪྦрǔːϰԐҸ䀢ਂc$Ђᙻїƿǿο开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ἀ开敲獬ⸯ敲獬䭐ȁ-!翅Цщȇ牤⽳潤湷敲浸偬Ջ̀̀뜀ༀჰ쀀琀섀ༀഀ矰鼀ЏЀꠀ̏㔵ྡ2ࠀg︀ЀCЀྪྦрǔːϰԐиł䀣ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃ͺශиł䀤ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃ඍӇඍශиł䀥ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃ჀͺჀශиł䀦ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃ӇჀӇиł䀧ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃[ܿᓿܿиł䀨ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃[ࢌᓿࢌиł䀩ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃[ᓿиł䀪ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃[ଧᓿଧиł䀫ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃[ᓿиł䀬ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃[ͺ[ශиł䀭ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃ᓿͺᓿශиł䀮ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃[ͺᓿͺиł䀯ਂc$їſƿNj㆜ǿο#Ϝǿ@쎩ϐ䭐Ѓ!쯶оƅ开敲獬ⸯ敲獬콬櫁ッ،ﯠ瑠鑟僮裆寓힡㻒놀쒕똬撌柭橺軇䤿괧⍦죑싙僁ᳶ鹢尭윯ྷ娰ᙌꑮᡰ彟㎆꺭邶넮榨噊䬋儓䋽槉藇魲ⱌ헉쩖얌⿹ğ牟汥⽳爮汥偳ŋⴂ᐀ࠀ℀蘀콜ༀ܀搀獲搯睯牮癥砮汭䭐·̃[ශᓿශĤ䀰ГdǯЂᙻїƁࠀƿǀࠀǿȁࠀȿ̿쎀οRectangle 183ญ}ࡪཾђྟྨ倍⠠⁔††䌠 ‽⁐吨 ⁐䌨㨩ྡ*ကPႁ•PྪЉЉྦшǔːϰԐŐ䀱гpǯЂᙻ…‡їƁࠀƿǀࠀǿȁࠀȿ̿쎀οRectangle 185ߵ ߵF Type of positionGender
Total
Managerial
8
3
11
Professional
31
13
44
Technical
52
17
69
Clerical
9
2.7
31
Total
100
55
155
P (T C) = P (T) + P (C):
Probability Distribution
Defined: It is the distribution of all possible outcomes of a particular event. Examples of probability distribution are: the binomial distribution (only 2 statistically independent outcomes are possible on each attempt) (Example coin flip) the normal distribution other underlying distributions exist (such as the Poisson, t, f, chi-square, ect.) that are used to make statistical inferences.
The normal probability distribution
The normal curve is bell-shaped that has a single peak at the exact centre of the distribution.The arithmetic mean, median, and mode of the distribution are equal and located at the peak The normal probability distribution is symmetrical about its mean (of the observations are above the mean and are below).It is determined by 2 quantities: the mean and the SD. The random variable has an infinite theoretical range (Tails do not touch X – axis). The total area under the curve is = 1Figure
68% of the area under the carve is between 1 SD 95% of the area under the carve is between 1.96 SD 99% of the area under the carve is between 2.58 SD Why the normal distribution is important? A/ Because many types of data that are of interest have a normal distributionCentral Limit theorem
sampling distribution of means becomes normal as N increases, regardless of shape of original distribution Binominal distribution becomes normal as N increases N.B: Normal distribution is a continuous one Binomial distribution is a quantitative discreteStandard normal distribution (curve)
A normal distribution with a of zero and SD of 1 is called standard normal distribution Any normal distribution can be converted to the standard normal distribution using the Z-statistics (value) Z-value (SND): is the distance between the selected value, designated X, and the population mean (M), divided by the population SD ( ) Z = The standard normal distribution curve is bell-shaped curve centered around zero with a SD=1Z- score
Z-score is often called the standardized value or Standard Normal Deviate (SND). It denotes the number of SD.s a data value X is distant from the and in which direction. A data value less than sample mean will have a z-score less then zero; A data value greater than the sample will have a z-score greater than zero; and A data value = the will have a z-score of zero
Normal curve table
The normal curve table gives the precise percentage of scores (values) between the (z-score of zero) and any other z-score. It can be used to determine:proportion of scores above or below a particular z-score proportion of scores between the and a particular z–score proportion of scores between two z–scoresBy converting raw scores to z-scores, can be used in the same way for raw sources. Can also used in the opposite way: Determine a z-score for a particular proportion of scores under the normal curve. * Table lists positive z-scores * Can work for negatives too * Why? Because curve is symmetrical
Steps for figuring percentage above or below a z-score:
Convert raw score to z-score, if necessary Draw a normal curve: - indicate where z-score falls - Shade area you are trying to find Find the exact percentage with normal curve tableFigure
Steps for figuring a z-score or raw score from a percentage:Draw normal curve, shedding an approximate area for the percentage concerned Find the exact z-score using normal curve table Convert z–score to raw score, if desired
Figure
Example: For = 2200, M = 2000, = 200, Z = (2200-2000)/200=1For = 1700, M = 2000, = 200, Z = (1700 – 2000)/200= -1.5 A z-value of 1 indicates that the value of 2200 is 1 SD above the of 2000, while a z-value of -1.5 indicates that the value of 1700 is 1.5 SD below the of 2000. Example: For M= 500, = 365, determine the position of 722 in SD units
Figure
= = = 0.61We can also determine how much of the area under the normal curve is found between any point on the curve and the Once you have a z-score, you can use the table to find the area of the z-score 0.61 (from table A) = 0 .2291 = 0.23 Therefore, 22.9% or 23%
Q/ How much of the population lies between 500 and 722?A/ 0.5 – 0.23 = 0.27Q/ How much of the population is to the left?A/ 0.5 + 0.23 = 0.73
Example: The daily water usage per person in an area, is normally distributed with a of 20 gallons and a SD of 5 gallons Q1/ About 68% of the daily water usage per person in this area lies between what 2 values? A/ About 68% of the daily water usage will lie between 15 and 25 gallons Q2/ What is the probability that a person from this area, selected at random, will use less then 20 gallons par day? A/ P (X < 20) = 0.5
Q3/ What percent uses between 20 and 24 gallons?The z-value associated with X=24: z = (24 -20)/ 5 = 0.8 From the table, the probability of z= 0.8 is 0.2119. Thus, P (20 < Ч < 24) = 0.5 – 0.2119 = 0.2881 = 28.81%
Figure
What percent of the population uses between 18 and 26 gallous?A/ The z-value associated with X = 18: z = (18-20)/5= -0.4and for X=26: z= (26-20)/5 = 1.2Thus P (18 <Ч < 26) = P (-0.4 < Z < 1.2) =0.6554 – 0.1151 =0.5403Example: Height of young women:The distribution of heights of women, aged 20-29 years, is approximately normal with =64 inch and SD= 2.7 inchQ/ Approximately, 68% of women have height between ……………. and ………….Q/ ~ 2.5% of women are shorter than ……..Q/ Approximately, what proportion of women are taller then 72.1=?