Lecture notes Statistics The Practice of Statistics for Business and Economics, ISBN: 9781319154127

Statistics lectures:

WEEK1:
CH4; probability:

Chapter 4.1; probability: the study of randomness

A random phenomenon has outcomes that we cannot predict with certainty but that have a regular
distribution in very many repetitions. We cannot predict.

The probability of an event is the proportion of times the event occurs in may repeated trials of a
random phenomenon.
➔ Probability describes only what happens in the long run.

Independence: if the outcome of one trial don’t influence the outcome of any other.
➔ Replacement: independent.
𝑃(𝐵|𝐴) = 𝑃(𝐵)

Disjoint: something happening exclude something else happening. (mutually exclusive).
➔ Disjoint events cannot be independent.

4.2; probability models

Probability model; a description of a random phenomenon in the language of mathematics.

When we toss a coin. The description of coin tossing has 2 parts:
1. A list of possible outcomes. (Sample space)
2. A probability for each outcome. (Assignment of probabilities P)

Sample spaces (S); set of list of all possible outcomes of a random process.

𝑆 = {ℎ𝑒𝑎𝑑𝑠, 𝑡𝑎𝑖𝑙𝑠}

Event; an outcome or a set of outcomes of a random phenomenon. An event is a subset of the
sample space.

Probability rules;
1.
0 ≤ 𝑃(𝐴) ≤ 1
2. All possible outcomes of the sample space together must have probability 1.

3. The additional rule for disjoint events:
The additional rule holds if A and B are disjoint but not otherwise.

𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)

→ 𝑡ℎ𝑖𝑠 𝑟𝑢𝑙𝑒 𝑒𝑥𝑡𝑒𝑛𝑑𝑠: 𝑃(𝐴 𝑜𝑟 𝐵 𝑜𝑟 𝐶) = 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶)

4. The complement rule: the probability that an event does not occur is 1 minus the probability
that the event does occur.
𝑃(𝑛𝑜𝑡 𝐴) = 𝑃(𝐴𝑐 ) = 1 − 𝑃(𝐴)

, 5. Multiplication rule for independent events: This multiplication rule holds if A and B are
independent but not otherwise.
➔ Only when independent.!

𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) 𝑃(𝐵)
Venn diagram showing disjoint events:




Venn diagram showing the complement Ac of an event A:
➔ The complement consists of all outcomes that are not in A:




Benford’s law;
What is the probability that the leftmost digit (“first digit’’) of a multidigit financial number is 9?
Many of us would assume the probability to be 1/9. Surprisingly, this is often not the case for
legitimately reported financial numbers. It is a striking fact that the first digits of numbers in
legitimate records often follow a distribution known as Benford’s law.




Equally likely outcomes: If a random phenomenon has k possible outcomes, all equally likely, then
each individual outcome has probability 1/k.




➔ If event A is independent, Ac is also independent, and Ac is independent of B.

0.2% (0.002)of the results are false positive.
If all of 150 are free of illegal drugs. The probability of a negative result is 1 – 0.02 = 0.998
The probability of at least one false-positive among the 150 people tested is:

𝑃(𝑎𝑡 𝑙𝑒𝑎𝑠𝑡 1 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒) = 1 − 𝑃(𝑛𝑜 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒)
= 1 − 𝑃(150 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒𝑠)

, = 1 − 0.998150
= 1 − 0.741 = 0.259
4.3; General probability rules

The union of any collection of events is the event that at least one of the collections occurs.
i.e. P (A or B)
➔ For two events A and B, the union is
the event {A or B} that A or B or both
occur.

when the events are joint:

𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) 𝑃(𝐵)
Above only when independent




Conditional probability P(B|A): gives the probability of one event under the condition that we know
another event has occurred. (requirement: if denominator > 0)

➔ Probability that B occurs, given the information that A occurs.
➔ You can read the bar u as “given the information that.
➔ Not-Independent events.

𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃(𝐴|𝐵) =
𝑃(𝐵)
Or:
𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃(𝐵|𝐴) =
𝑃(𝐴)

Multiplication rule:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) 𝑃(𝐵|𝐴)

𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) 𝑃(𝐵)

Example multiplication rule:




The intersection of any collection of events is the event that all the events occur.
➔ Contains all outcomes that are in both A and B, but not outcomes in A alone or B alone.

Example:
𝑃(𝐴 𝑎𝑛𝑑 𝐵 𝑎𝑛𝑑 𝐶) = 𝑃(𝐴) 𝑃(𝐵|𝐴) 𝑃(𝐶|𝐴 𝑎𝑛𝑑 𝐵)

, Tree diagrams:

𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) 𝑃(𝐵|𝐴)

𝑃(𝐴𝑐 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴𝑐 ) 𝑃(𝐵|𝐴𝑐 )

𝑃(𝐵) = 𝑃(𝐴 𝑎𝑛𝑑 𝐵) + 𝑃(𝐴𝑐 𝑎𝑛𝑑 𝐵)

𝑃(𝐵) = 𝑃(𝐴) 𝑃(𝐵|𝐴) + 𝑃(𝐴𝑐 ) 𝑃(𝐵|𝐴𝑐 )




Bayes’s rule:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) 𝑃(𝐵|𝐴)

𝑃(𝐴 𝑎𝑛𝑑 𝐵)
𝑃(𝐴|𝐵) =
𝑃(𝐵)

𝑃(𝐵) = 𝑃(𝐴) 𝑃(𝐵|𝐴) + 𝑃(𝐴𝑐 ) 𝑃(𝐵|𝐴𝑐 )

Combining these facts:

𝑃(𝐴)𝑃(𝐵|𝐴)
𝑃(𝐴|𝐵) = 𝑃(𝐴) 𝑃(𝐵 𝐴
| )+𝑃(𝐴𝑐 ) 𝑃(𝐵|𝐴𝑐 )



Independence rule:
𝑃(𝐵|𝐴) = 𝑃(𝐵)

Summary general rules chapter 4.2:
Additional rule:
➔ When they are joint.
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
Multiplication rule:
➔ When they are not independent.
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) 𝑃(𝐵|𝐴)