Theoretical and Empirical Probabilities
 Topic Index | Algebra2/Trig Index | Regents Exam Prep Center


You are already familiar with a plethora (a bunch) of information about working with probability.   Let's quickly refresh our memories:
(Additional review can be found at the Algebra web site under Probability.)

Probability describes the chance that an uncertain event will occur.
Probability of success is defined as:  
Probabilities range from 0 to 1 (where 0 means "will never occur" and 1 means "absolutely certain to occur").
Empirical Probability of an event is an "estimate" that the event will happen based on how often the event occurs after collecting data or running an experiment (in a large number of trials).  It is based specifically on direct observations or experiences.
Formula for probability of event E: 
Theoretical Probability of an event is the number of ways that the event can occur, divided by the total number of outcomes.  It is finding the probability of events that come from a sample space of known equally likely outcomes.
Formula for probability of event E (from sample space S): 
"AND" for independent events A and B:
   
    

Complement of event A (denoted A'):
          

"OR" for not mutually exclusive  events A and B:

 
Mutually exclusive events A and B (no outcomes in common):

also written
 Conditional probability of event B given event A,  for dependent events A and B:
 Remember to read carefully when dealing with dependent compound events for references to "with replacement" or "without replacement."


Here are some warm-up examples:
(Decimal answers will be rounded to the nearest tenth, when needed.)

1.  At a school fair, the spinner represented in the accompanying diagram is spun twice.  What is the probability that it will land in section G the first time and then in section B the second time?

Solution:  The right angle tells us that sections R and G are each 1/4 of the entire circle, with section B being 1/2 of the circle. 
Answer:  
 

 

2.  Shandra and Alexi roll two dice 50 times and record their results in the accompanying chart.
a.)  What is their empirical probability of rolling a 7?
b.)  What is the theoretical probability of rolling a 7?
c.)  How do the empirical and theoretical probabilities compare?
 
 
Sum of the rolls of two dice
3, 5, 5, 4, 6, 7, 7, 5, 9, 10,
12, 9, 6, 5, 7, 8,  7, 4, 11, 6,
8, 8, 10, 6, 7, 4, 4, 5, 7, 9,
9, 7, 8, 11, 6, 5, 4, 7, 7, 4,
3, 6, 7, 7, 7, 8, 6, 7, 8, 9
 
Solution:  
a.)  Empirical probability (experimental probability or observed probability) is 13/50 = 26%.
b.)  Theoretical probability (based upon what is possible when working with two dice) = 6/36 = 1/6 = 16.7%  (check out the table at the right of possible sums when rolling two dice).
c.)  Shandra and Alexi rolled more 7's than would be expected theoretically.

 

3.  The accompanying figure is a square.  The interior sections are formed using congruent squares.  If this figure is used as a dart board, what is the probability that the dart will hit the shaded blue region?

Solution:  The large square is broken into 9 smaller congruent squares of which 5 are shaded blue.  The probability is 5/9 = 55.6%.

 


 

4.  Two colored dice (one red, one white) are rolled.
a.)  What is the probability of rolling "box cars" (two sixes)?
b.)  What is the probability of rolling "box cars" knowing the first toss is a six?

Solution: 
a.  The probability of getting "box cars" (two sixes) is (1/6)•(1/6) = 1/36. 

 
b.  If, however, we roll the dice and see that the white die shows a six (and the red die is out of sight), the probability of the red die being six is 1/6.  The probability of rolling "box cars", knowing that the first roll is a six, is 1/6.  The probability changes when you have partial information about the situation.   This is a conditional probability situation.

 
 


5.  A pair of dice are rolled.  What is the probability of rolling 10 or less?

Solution:  The complement of rolling "10 or less" is rolling 11 or 12.
P(10 or less) = 1 - P(11 or 12) = 1 - [P(11) + P(12)] = 1 - (2/36 + 1/36) = 33/36 = 11/12
(refer to the chart in question 2 to see the number of occurrences of rolling an 11 or a 12)