Theoretical Probability
Theoretical Probability is using mathematics and reasoning to determine the probability of an event without doing experiments or simulations. In experiments, actual data is used to determine the relative frequency (probability) of an event.
Determining the likelihood of an event using theoretical probability is faster than performing experiments.
The probability of some simple events (the event has only one outcome) is already known.
Example:
- Rolling a 5 with a single die.
- Pulling a spade from a regular deck of cards. How many ways (outcomes) can you pull a spade? How many ways can you pull any card from the deck?
P(Drawing a Spade) =
Note: P(event) is notation for ‘the probability of the event occurring.
- Rolling a prime number with a regular die.
When determining probability of an event, ask yourself:
1How many outcomes are possible? Are all outcomes equally likely (probable)? For example is each side of the coin equally likely? Is each face of the die equally likely? Is each card in the deck of cards equally likely?
If a bag contains 5 yellow and 2 green marbles, is each colour equally likely to be drawn?
2If you want the probability of an event, count how many ways the event can occur? (how many outcomes).
For example:
a)rolling an even number with a die
b)drawing a seven from a regular deck of cards.
3
Does this method work for these examples?
Rolling a 5 on a die?
Drawing a spade from a regular deck?
Rolling a prime number with a regular die?
Drawing a 7 from a deck of cards?
Drawing a black face card from a deck of cards?
Sample space - a collection of all possible outcomes of the experiment.
Event space – the collection of all outcomes of an experiment that correspond to a particular event.
Theoretical Probability:
A is the collection of outcomes for the event. (The Event Space)
S is all possible outcomes for the experiment. (TheSample Space)
n( ) means the number of elements or in this case outcomes in that set.
A Venn diagram can be used to model the relationship between the outcomes of event A and the sample space S.
Determine the Probability of each Event:
1A bag contains 3 red marbles, 5 green marbles and 7 gold marbles. What is the probability of drawing red marble?
What is the probability of drawing a gold marble?
2Drawing a face card from a deck of cards.
3Drawing any card but a face card from the deck.
What did you notice about your answers to #2 and #3?
What was the sum of the probabilities in #2 and #3?
Example #2 and #3 are called complementary events.
If A represents an event then represents the complement of A.
are all the events in S that are not in A. This means all outcomes of the experiment that are not in A.
A Venn diagram can be used to demonstrate
Also as we noticed in #2 and #3.
This is often an easier way to calculate probability.
Examples:
1.Find the probability of selecting a number from 1 to 100 that is not divisible by 15.
P(Number not divisible by 15) =
- In a bag with 25 marbles 4 are blue. What is the probability of drawing a marble that is not blue?
Homework Page 218 # 1ac, 3, 4, 5e, 6b, 8, 10, 12, 13,17