Question

a six - sided number cube is rolled and the proportion of 3s is calculated. the graph shows the long - run relative frequency of 100 rolls of this die. which conclusion can be drawn from this graph?
there is a 20% chance of rolling a 3 with this die.
the long - run frequency of rolling a 3 appears to be about 0.17.
because the probabilities on the left part of the graph fluctuate so much, we cannot conclude that this is a fair number cube.
this number cube must be fair because each outcome of rolling a 1, 2, 3, 4, 5, or 6 has the same probability of occurring.

Explanation:

Step1: Analyze the first option

A fair six - sided die has a probability of \(\frac{1}{6}\approx0.167\) of rolling a 3. 20% is \(0.2\), which is not equal to \(\frac{1}{6}\), so the first option is wrong.

Step2: Analyze the second option

Looking at the graph, as the number of rolls (frequency) increases, the proportion of 3s (probability) stabilizes around \(0.17\). This is consistent with the long - run relative frequency approach.

Step3: Analyze the third option

Just because the left - hand side (with fewer trials) has fluctuations does not mean we cannot conclude it is a fair die. The long - run relative frequency (as the number of trials gets large) is what is important for determining fairness.

Step4: Analyze the fourth option

We cannot say the die “must be fair” just because we assume each outcome has the same probability. The graph is used to estimate the probability from data (empirical probability), and we should base our conclusion on the graph.

Answer:

The long - run frequency of rolling a 3 appears to be about 0.17.