The primary objective of this section is to use the basic definition of probability in finding theoretical and experimental probabilities. The basic definition of probability is the number of successful outcomes
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P =
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number of possible outcomes
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total number of possibilities
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This can be written as P(E) = n(E) where P(E) means n(S) the probability of event E happening, n(E) means the number of times E could happen and n(S) means the number in the sample set which is the total number of possible outcomes.
Consider a die. It has six surfaces, and each surface has a set of 1, 2, 3, 4, 5, or 6 dots on it. If I roll a die, the only possible outcomes are 1, 2, 3,4 , 5 or 6. These six elements make up the sample set for our event the rolling of the die.
If I roll a die, I can ask for the probability of different events happening. What is the probability of the following.
a. P(1) = ___
b. P(even number) = ___
c. P(8) = ___
d. P(n > 5) = ___ where n means the number on the die
e. P(odd number) = ___
f. P(n 7) = ___
Since each of these is answered by P(E) = n(E)/n(S), the n(S) answers are as follows.
(figure available in print form)
Notice O means no possibility the event will happen. 1 means it will always happen. The probability of an event will always be between 0 and 1 or equal to one of them.
(figure available in print form)
If I roll a die, P(2) = 1/6. This could be written 1/6 as 1 or as 0.166. P( n 4) = 1/2 or 0.5. This may be an interesting way to review students’ basic skills in fractions and decimals. We’ll use both below.
I want to roll a die 12 times to see if the probability of getting 4 really is 1/6 as indicated by the definition.
Theoretical probability is what we have been talking about up to this point. Now we want to move out of the theoretical into the real world and try out that probability with a real die. I’ll now roll a die 12 times.
results: theoretical probability P(4) = 1/6
experimental probability EP(4) = 4/12 = 1/3
In class one student could roll the die, another could tally it on the board. If we’re lucky there will be a discrepancy to point out the difference between theoretical and experimental probability with a small sample of 12.
At this point, letting students roll dice and get how many times 4 comes up for each of them could be organized as follows.
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Times Roll Die
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Number of 4’s
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P(fraction)
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P(decimal)
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12
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___
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___
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20
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___
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___
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30
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___
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___
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For each of the above three experiments, have students calculate the experimental probabilities they find using both fractional and decimal forms.
Notice that by using the basic definition of probability we can find simple probabilities, both theoretical, the probabilities that you might expect, and the experimental, the probabilities you get in the real world by doing experiments like roll a die, flip a coin, or draw a card from a deck. Further, by using small samples the experimental probability might be quite different from the theoretical, but as we increase the number of tries, that is as the number in our sample increases, the experimental probability moves closer and closer to the theoretical. How large a sample is needed? Again, 30 is usually considered to be fairly reliable sample.
Other easy activities done in the same or similar way are to ask how the theoretical and experimental probabilities compare for P(n > 1) in rolling a die, or for P(T) the probability of getting tails when flipping a coin, or P(2) the probability of drawing a two from a pack of cards.
In summary, with simple probability problems we can use the basic definition of probability to experiment with the difference between theoretical and experimental probabilities. The “simple” here means problems where it is easy to count the numbers you need as opposed to more difficult probability problems where the basic idea is the same but the counting of needed numbers becomes more difficult.
