QUESTION 6
Easy
Suppose you roll a 6-sided die 12 times and get a '3' three times.
(i) What is the experimental probability of rolling a '3'?
(ii) What is the theoretical probability of rolling a die?
(iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?
SOLUTION
1
Note down what is given
Total rolls \(= 12\)
Number of times '3' appeared \(= 3\)
Number of times '3' appeared \(= 3\)
2
Find the experimental probability of rolling a '3'
\(P(3) = \frac{Number\,of\, times\, 3\, occurs}{Total\, rolls}\)
⇒ \(P(3) = \frac{3}{12} = \frac{1}{4} = 0.25\)
Experimental probability of rolling a '3' \(= \frac{1}{4}\) or \(0.25\).
⇒ \(P(3) = \frac{3}{12} = \frac{1}{4} = 0.25\)
Experimental probability of rolling a '3' \(= \frac{1}{4}\) or \(0.25\).
3
Find the theoretical probability of rolling a '3'
Number of favourable outcomes \(= 1\), since only one face of the die shows '3'.
A die has 6 faces, so
Total outcomes \(= 6\)
\(P(3) = \frac{Number\, of\, favourable\, outcomes}{Total\, possible\, outcomes}\)
⇒ \(P(3) = \frac{1}{6}\) ≈ \(0.167\)
Theoretical probability of rolling a '3' \(= \frac{1}{6}\) or approximately \(0.167\).
A die has 6 faces, so
Total outcomes \(= 6\)
\(P(3) = \frac{Number\, of\, favourable\, outcomes}{Total\, possible\, outcomes}\)
⇒ \(P(3) = \frac{1}{6}\) ≈ \(0.167\)
Theoretical probability of rolling a '3' \(= \frac{1}{6}\) or approximately \(0.167\).
4
Difference in the two probabilities
The experimental probability is based on only 12 rolls. In a small number of trials, results can vary due to chance.
The two probabilities are different because the experiment was done only a few times.
The two probabilities are different because the experiment was done only a few times.
5
Discuss case in which the die is rolled many more times
If the die is rolled 60 times, 600 times, or 6000 times,
the experimental probability will usually get closer and closer to the theoretical probability.
the experimental probability will usually get closer and closer to the theoretical probability.
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Final Answer :
Experimental probability of rolling a '3' \(= \frac{1}{4}\) or \(0.25\).
Theoretical probability of rolling a '3' \(= \frac{1}{6}\) or approximately \(0.167\).
Concept Note
Experimental Probability is based on actual experiments or observations. It is given by the formula:
\(P(E) = \frac{Number\, of\, times\, the\, event\, occurs}{Total\, number\, of\, trials}\)
Theoretical Probability is based on what should happen mathematically. It is given by the formula:
\(P(E) = \frac{Number\, of\, favourable\, outcomes}{Total\, possible\, outcomes}\)
For a fair 6-sided die,
Possible outcomes\( = {1, 2, 3, 4, 5, 6}\), with each number having an equal chance of occuring.