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Theoretical Probability Solutions

• Probability is a    measure    of how likely an event is to occur.

  • It is always a number    between 0 and 1   .

  • Depending on the problem, probability will be written as:
 
    •   a fraction (ratio)   
    •   a decimal   
 
    •   a percentage   
• A probability of zero (0%) means the event will   never happen   .

• A probability of one (100%) means the event will   always happen   .

Vocabulary	Example
The   sample space    is a list of all possible outcomes. Note The total number of equally likely outcomes in the sample space	Sides of a number cube: {1, 2, 3, 4, 5, 6} Note 6 (the cube has 6-sides)
An   event    is the favorable result or set of favorable results of a probability experiment. P(E)	Rolling an even number on a number cube P(even)
An   outcome    is the result of a probability experiment.	Rolling a 2 on a number cube
A   complement    to an event is all of the events that are not the event. $\mathrm{P}(\mathrm{not} \mathrm{E})=1–\mathrm{P}\left(\mathrm{E}\right)$ or P(E’)	The odd numbers are a complement to the evens on a number cube. P(not even) or P(odd)

Note

In mathematics, unless you are completing an experiment or activity, you are calculating the theoretical probability.

• All the events in the sample space have   an equal chance of occurring   .
 
•   Theoretical probability    is the expected probability using math and logical reasoning.

• Theoretical probability is the   ratio    of favored, or desired, outcomes to the total number of possible outcomes.


$\mathrm{P}\left(\mathrm{favorable} \mathrm{event}\right)=\frac{\mathrm{favorable} \mathrm{outcome}}{\mathrm{total} \mathrm{possible} \mathrm{outcomes}}$

Example 1

A bag contains seven red, eight yellow, five blue, six white, and four purple marbles. Determine the probability.

Total: $7+8+5+6+4=30$

Example 2

A security system requires a 5-digit code with no repeating numbers. If the digits 0 through 9 are allowed, what is the probability that a randomly generated code will contain the digits 1, 2, and 3 in that exact order, but the other two digits can be any other numbers (also in any order)?

Total number of codes: $nPr(10, 5)=30,240$

Codes allowed:

$\frac{1}{\mathrm{must} \mathrm{be} \mathrm{a} 1} \frac{1}{\mathrm{must} \mathrm{be} \mathrm{a} 2} \frac{1}{\mathrm{must} \mathrm{be} \mathrm{a} 3} \frac{7}{\mathrm{not} 1, 2, 3} \frac{6}{\mathrm{not} 1, 2, 3 ?} = 42 \mathrm{Possibilities}$

$\mathrm{P}(\mathrm{codes} \mathrm{starting} \mathrm{with} 1, 2, 3)=\frac{42}{30 ,240}=\frac{1}{720}$

Example 3

The diagram shows all possible pairs when two dice are rolled. Find the probability of each outcome to the nearest whole percent. 

4. Explain which event is most likely to occur from problems A through C.
   
Rolling   “not doubles”    is the event most likely to occur because   the probability is 83%   , which is closest to 100%.