The probability of an event is a value that quantifies the possibility that the event will occur.
The value of any probability is always between |0| and |1|, inclusive. This value can be expressed as a percentage, fraction or decimal number.
There are different types of probabilities.
The theoretical probability of an event is a probability determined using mathematical reasoning.
When the elementary events of a random experiment are equiprobable, we can calculate the theoretical probability as follows:
||\text{Theoretical Probability} = \dfrac{\text{Number of Favourable Outcomes}}{\text{Number of Possible Outcomes}}||
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The probability of rolling a |3| when throwing a |6|-sided die is |1| out of |6,| since there is |1| favourable outcome (rolling the |3|) out of |6| possible outcomes (the |6| sides of the die). ||P(6)=\dfrac{1}{6}||
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The probability of getting an odd number when rolling a |6|-sided die is |1| out of |2,| since there are |3| favourable results (|1,| |3,| and |5|) out of |6| possible results (the |6| sides of the die). ||P(\text{rolling an odd number})=\dfrac{3}{6}=\dfrac{1}{2}||
The experimental probability of an event is determined using the results of an experiment.
The experimental probability of an event is often used when a theoretical probability is difficult, or even impossible, to calculate. To find it, the same random experiment must be repeated a large number of times. The greater the number of times the experiment is repeated, the more the experimental probability of the event is a good estimate of the theoretical probability.
The experimental probability of an event can be determined as follows:
||\text{Experimental Probability} = \dfrac{\begin{gather}\text{Number of Times the}\\ \text{Favourable Outcome Occurs }\end{gather}}{\begin{gather}\text{Number of Times the}\\\text{Experiment is Repeated}\end{gather}}||
A plastic cup is dropped on the floor. What is the probability that it will come to rest on its open base?
It is very difficult to calculate the theoretical probability of each event because the 3 elementary events are not necessarily equiprobable. To calculate the experimental probability of the desired outcome, the experiment must be repeated many times. Therefore, we can determine the number of times that the glass falls on its open base in relation to the number of times the experiment is repeated.
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Throw |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
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Result |
Side |
Side |
Closed base |
Side |
Side |
Side |
Open base |
Side |
Side |
Side |
Side |
Open base |
||\begin{align}P(\text{Open Base})&= \dfrac{\begin{gather}\text{Number of times that the}\\ \text{favorable outcome occurs}\end{gather}}{\begin{gather}\text{Number of times that the}\\\text{experiment is repeated}\end{gather}}\\&=\dfrac{2}{12}\\&=0.1\overline{6}\end{align}||
So, if we toss the glass one more time, we can assume that it has a |16.\overline{6}\ \%| chance of landing on its open base.
Even if one of the outcomes does not occur in the random experiment, we cannot conclude that this outcome is impossible.
When tossing a coin, what is the probability of the coin landing on heads?
Theoretical probability
If the coin is not rigged, then the |2| events (getting "heads" and getting "tails") are equiprobable.
||\begin{align}P(\text{get « heads »})&=\dfrac{1}{2}\\&=50\ \%\end{align}||
Experimental probability
If we suspect that the coin is rigged, we cannot calculate the theoretical probability. The coin is tossed |35| times. We get "heads" |16| times and “tails” |19| times.
||\begin{align}P(\text{get « heads »})&=\dfrac{16}{35}\\&\approx 45.7\ \%\end{align}||
The more we repeat the experiment, the closer we should approach the theoretical probability of |50\ \%.| If this is not the case, then the coin is indeed rigged.
The subjective probability of an event is a probability determined by personal judgment.
In some situations, there is not enough information to determine the theoretical or experimental probability of an event. We must then rely on our observations and our judgment to establish this probability. This is called subjective probability, since it can vary from one person to another.
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Since he feels he has studied hard, Julian estimates the probability of passing his next math exam to be |90\ \%.|
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By analyzing the statistics, we estimate that the Montreal Canadiens have a |50\ \%| chance of winning the Stanley Cup this year.
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The clouds are grey and the air is humid, so we estimate that there is an |80\ \%| chance of rain.