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KONSEP DASAR PROBABILITAS

BUDHI SETIAWANTEKNIK SIPIL UNSRI

STATISTIK DAN PROBABILISTIK

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Kondisi acak

Kondisi acak adalah satu kondisi dimana hasil atau keadaan tidak dapat diprediksi

Contoh: Status penyakit

Anda memiliki penyakitAnda tidak memiliki penyakit

Hasil test positifHasil test negatif

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Definisi Probabilitas

Probabilitas adalah nilai antara 0 dan 1 yang dituliskan dalam bentuk desimal ataupun pecahan.

Secara sederhana, Probability adalah bilangan antara 0 dan 1 yang menunjukkan suatu hasil yang diperoleh dari kondisi acak.

Untuk satu susunan kemungkinan yang lengkap dalam kondisi acak, maka total atau jumlah probabilitas adalah harus sama dengan 1.

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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.

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Contoh: Probabilitas dalam perencanaan transportasi

Probabilitas kejadian 5 mobil menunggu untuk berbelok kanan adalah 3/60 (2/60 + 1/60)

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

. . .

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 dan diperoleh 60 hasil pengamatan.

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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.

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Determining the Relative Frequency(Probability) of an Outcome

Method 2: Observe the Relative Frequency of random circumstancesThe 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.

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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.).

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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 n

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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)

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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.

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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.

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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 EventsIf 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.

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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.