STATISTIK DAN PROBABILISTIK KONSEP DASAR PROBABILITAS BUDHI SETIAWAN

STATISTIK DAN PROBABILISTIK KONSEP DASAR PROBABILITAS BUDHI SETIAWAN TEKNIK SIPIL UNSRI 1

Kondisi acak adalah satu kondisi dimana hasil atau keadaan tidak dapat diprediksi Contoh: Status penyakit Anda memiliki penyakit Anda tidak memiliki penyakit Hasil test positif Hasil test negatif 2

Definisi Probabilitas l Probabilitas adalah nilai antara 0 dan 1 yang dituliskan dalam bentuk desimal ataupun pecahan. l Secara sederhana, Probability adalah bilangan antara 0 dan 1 yang menunjukkan suatu hasil yang diperoleh dari kondisi acak. l Untuk satu susunan kemungkinan yang lengkap dalam kondisi acak, maka total atau jumlah probabilitas adalah harus sama dengan 1. 3

Assigning Probability How likely it is that a particular outcome will be the result of a random circumstance The Relative Frequency Interpretation of Probability In situations that we can imagine repeating many times, we define the probability of a specific outcome as the proportion of times it would occur over the long run -- called the relative frequency of that particular outcome. 4

Contoh: Probabilitas dalam perencanaan transportasi Di suatu ruas jalan direncanakan untuk membuat jalur khusus belok kanan. Probabilitas 5 mobil menunggu berbelok diperlukan untuk menentukan panjang garis pembagi jalan. Untuk keperluan ini dilakukan survey selama 2 bulan diperoleh 60 hasil pengamatan. Banyaknya Mobil Jumlah Pengamatan Frekuensi relative 0 4 4/60 1 16 16/60 2 20 20/60 3 14 14/60 4 3 3/60 5 2 2/60 6 1 1/60 7 0 0 8 0 0 . . . Probabilitas kejadian 5 mobil menunggu untuk berbelok kanan adalah 3/60 (2/60 + 1/60) 5

Determining the Relative Frequency (Probability) of an Outcome Method 1: Make an Assumption about the Physical World (there is no bias) A Simple Lottery Choose a three-digit number between 000 and 999. Player wins if his or her three-digit number is chosen. Suppose the 1000 possible 3 -digit numbers (000, 001, 002, 999) are equally likely. In long run, a player should win about 1 out of 1000 times. Probability = 0. 0001 of winning. This does not mean a player will win exactly once in every thousand plays. 6

Determining the Relative Frequency (Probability) of an Outcome Method 2: Observe the Relative Frequency of random circumstances The Probability of Lost Luggage “ 1 in 176 passengers on U. S. airline carriers will temporarily lose their luggage. ” This number is based on data collected over the long run. So the probability that a randomly selected passenger on a U. S. carrier will temporarily lose luggage is 1/176 or about 0. 006. 7

Proportions and Percentages as Probabilities Ways to express the relative frequency of lost luggage: • The proportion of passengers who lose their luggage is 1/176 or about 0. 006 (6 out of 1000). • About 0. 6% of passengers lose their luggage. • The probability that a randomly selected passenger will lose his/her luggage is about 0. 006. • The probability that you will lose your luggage is about 0. 006. Last statement is not exactly correct – your probability depends on other factors (how late you arrive at the airport, etc. ). 8

Estimating Probabilities from Observed Categorical Data Assuming data are representative, the probability of a particular outcome is estimated to be the relative frequency (proportion) with which that outcome was observed. Approximate margin of error for the estimated probability is 9

Nightlights and Myopia Assuming these data are representative of a larger population, what is the approximate probability that someone from that population who sleeps with a nightlight in early childhood will develop some degree of myopia? Note: 72 + 7 = 79 of the 232 nightlight users developed some degree of myopia. So we estimate the probability to be 79/232 = 0. 34. This estimate is based on a sample of 232 people with a margin of error of about 0. 066 (1/√ 232 = ± 0. 666) 10

The Personal Probability Interpretation Personal probability of an event = the degree to which a given individual believes the event will happen. Sometimes subjective probability used because the degree of belief may be different for each individual. Restrictions on personal probabilities: • Must fall between 0 and 1 (or between 0 and 100%). • Must be coherent. 11

Probability Definitions and Relationships Sample space: collection of unique, nonoverlapping possible outcomes of a random circumstance. Simple event: one outcome in the sample space; a possible outcome of a random circumstance. Event: a collection of one or more simple events in the sample space; often written as A, B, C, and so on. 12

Assigning Probabilities to Simple Events P(A) = probability of the event A Conditions for Valid Probabilities 1. Each probability is between 0 and 1. 2. The sum of the probabilities over all possible simple events is 1. Equally Likely Simple Events If there are k simple events in the sample space and they are all equally likely, then the probability of the occurrence of each one is 1/k. 13

Example: Probability of Simple Events Random Circumstance: A three-digit winning lottery number is selected. Sample Space: {000, 001, 002, 003, . . . , 997, 998, 999}. There are 1000 simple events. Probabilities for Simple Event: Probability any specific three-digit number is a winner is 1/1000. Assume all three-digit numbers are equally likely. Event A = last digit is a 9 = {009, 019, . . . , 999}. Since one out of ten numbers in set, P(A) = 1/10. Event B = three digits are all the same = {000, 111, 222, 333, 444, 555, 666, 777, 888, 999}. Since event B contains 10 events, P(B) = 10/1000 = 1/100. 14