Statistics for Decision Makers - 05.01 - Probability

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title

05.01 - Probability

author

Bernard Szlachta (NobleProg Ltd) bs@nobleprog.co.uk

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Inferential Statistics and the problem with business examples。

If you choose an answer to this question at random, what is the chance you will be correct?

1. 25%
2. 50%
3. 60%
4. 25%

Inferential Statistics。

• Inferential statistics is built on the foundation of probability theory
• The idea of probability has been plagued by controversy from the beginning of the subject to the present day

Approaches to probability

1. Symmetrical Outcomes
2. Frequentist
3. Subjective

Symmetrical Outcomes。

If there are N symmetrical outcomes, the probability of any given one of them occurring is taken to be 1/N.

Fair Coin	Dice
The two possible outcomes of tossing a fair coin seem not to be distinguishable in any way that affects which side will land up or down The probability of heads is taken to be 1/2, as is the probability of tails	If a six-sided die is rolled, the probability of any one of the six sides coming up is 1/6

Relative Frequencies。

• If we tossed a coin millions of times, we would expect the proportion of tosses that came up heads to be pretty close to 1/2
• As the number of tosses increases, the proportion of heads approaches 1/2
• Therefore, we can say that the probability of a head is 1/2

Example。

If it has rained in Seattle on 62% of the last 100,000 days, then the probability of it raining tomorrow might be taken to be 0.62

...Is that correct?

Weather in Seattle。

• This is a natural idea but nonetheless unreasonable if we have further information relevant to whether it will rain tomorrow
• For example, if tomorrow is August 1, a day of the year on which it seldom rains in Seattle, we should only consider the percentage of the time it rained on August 1
• But even this is not enough since the probability of rain on the next August 1 depends on the humidity (the chances are higher in the presence of high humidity)
• So, we should consult only the prior occurrences of August 1 that had the same humidity as the next occurrence of August 1
• Of course, wind direction also affects probability ... You can see that our sample of prior cases will soon be reduced to an empty set
• Past meteorological history is misleading if the climate is changing

Frequentist Approach to Probability。

• Like most work in the field, the present text adopts the frequentist approach to probability in most cases
• Moreover, almost all the probabilities we shall encounter will be nondogmatic, that is, neither 0 nor 1
• An event with probability 0 has no chance of occurring; an event with probability 1 is certain to occur
• It is hard to think of any examples of interest to statistics in which the probability is either 0 or 1
• Even the probability that the sun will come up tomorrow is less than 1

Example。

• The weather forecaster says "there is a 10% chance of rain"
• You decide to have a picnic outdoors and, it rains
• Was the weather person wrong?

Blame the forecaster。

• No, she did not say it would not rain, only that rain was unlikely
• She would have been wrong only if she said that the probability was 0 and it subsequently rained
• However, if you kept track of her weather predictions over a long period of time and found that it rained on 50% of the days that the weather person said the probability was 0.10, you could say her probability assessments are wrong
• According to our frequency interpretation, it means that it will rain 10% of the days on which rain is forecast with this probability

Subjective Probability 。

• Outside of scientific interest

Quiz。

Please find the Quiz here

Quiz

1 Select all that apply. Probability can be thought of as:

	Symmetrical outcomes
	Relative frequencies
	Subjective

Answer >>

symmetrical outcomes, relative frequencies, subjective

There is a debate about how to interpret probability. All three of these can be ways to think of probability.

2 The paper says there is an 80% chance of rain today, so you plan indoor activities. Then it doesn't rain. Was the forecast wrong?

	Yes
	No

Answer >>

No

The forecast would have only been wrong if it had predicted a 100% chance of rain. However, the probability assessments are wrong if, over time, it usually doesn't rain on days where an 80% chance of rain is predicted.

3 Most probabilities we will deal with in psychology are zero or one.

	True
	False

Answer >>

False

We will deal with very few probabilities that are zero (definitely will not happen) or one (definitely will happen).

4 When would you say that the forecaster is wrong?

	When the weather forecast is incorrect 10 days in a row.
	1 in 20 of forecasts are incorrect.
	If 80 out of 100 forecasts with 10% chances of rain were sunny.