Classical Approach: Based on Equally Likely Events

$$\text{Probability of an Event X}=\frac{N_x}{N}=\frac{\text{Number of Outcomes That Satisfy That Event}}{\text{Total Number of Possible Outcomes}}$$

This approach assumes all outcomes are equally likely to occur. If an experiment has n possible outcomes, this method would assign a probability of 1/n to each outcome. It is necessary to determine the number of possible outcomes.

Experiment: Rolling a Die

$\text{Outcomes: }\{1,2,3,4,5,6\}\text{, }N=6$

• Probability that a face with five spots turns up when you roll:
  • $P\left(x=5\right)=\frac{1}{6}$
• Probability that you roll four or less:
  • $P\left(x<=4\right)=\frac{4}{6}=0.67\text{ or }67\%\text{ (in this case, there are four possible outcomes, 1, 2, 3, 4 that satisfy the condition 4 or less)}$
• Also note that the possibility you roll more than four:
  • $P\left(x>4\right)=1-0.67=0.33\text{ or }33\%$

 

Relative Frequency: Assigning Probabilities Based on Experimentation or Historical Data

$\text{Probability of an event }X=\frac{N_x}{N}=\frac{\text{Number of Outcomes That Satisfy That Event}}{\text{Total Number of Events in the Population}}$

You may remember the cumulative and relative frequencies we calculated for Google’s Project Fi customers. The relative frequencies we calculated for each category can be used as a probabilistic estimate for customer spending patterns. So, we can say there is a 6.5% chance that a new customer will spend between $30–$45, 60.5% chance that a new customer will spend $45 or less, and so on. In other words, we are using this data from a sample of 200 customers to predict the behavior of a new customer.

Table 2.4. Probability of Customer Spending

Spending amount (X) in dollars	Relative frequency P(X)	Cumulative relative frequency P(X<=...)
0–15	$\frac{71}{200}=0.355$	$\frac{71}{200}=0.355$
>15–30	$\frac{37}{200}=0.185$	$\frac{108}{200}=0.54$
>30–45	$\frac{13}{200}=0.065$	$\frac{121}{200}=0.605$
>45–60	$\frac{9}{200}=0.045$	$\frac{130}{200}=0.65$
>60–75	$\frac{10}{200}=0.05$	$\frac{140}{200}=0.7$
>75–90	$\frac{18}{200}=0.09$	$\frac{158}{200}=0.79$
>90–105	$\frac{28}{200}=0.14$	$\frac{186}{200}=0.93$
>105–120	$\frac{14}{200}=0.07$	$\frac{200}{200}=1$

 

Probability a new customer spends:

• Between $30–$45   $P\left(30<X<=45\right)=0.065\text{ or }6.5\%$
• $45 or less: $P\left(X<=45\right)=0.605\text{ or }60.5\%$
• More than $45 : $P\left(X>45\right)=1-P\left(X<=45\right)=1-0.605=0.395\text{ or }39\%$

The same information can also be displayed in a chart:

Example: What Is the Probability That a Family in Your County Has an Income > 60,000?

• Last census data shows that there were 54,345 households in your county, of which 31,496 had a income above 60K
  • $\text{Probability of event X (i.e. family income }>60K)=P\left(X>60K\right)=\frac{31,496}{54,345}=.580$
• Sales Management Magazine reports 55,100 households, with 32,047 having income > 60K
  • $\text{Probability of event X (i.e. family income }>60K)=P\left(X>60K\right)=\frac{32,047}{55,100}=.5$

 

 

 

Subjective Approach: Assigning Probabilities Based on the Assignor’s (Subjective) Judgment

In the subjective approach, we define probability as the degree of belief that we hold in the occurrence of an event.

For example, based on historical performance and current market conditions, there is a 75% chance that stock price will go up in the next quarter.

Subjective probabilities have personal biases and may vary widely from person to person.