Question

number cube
number rolled | 1 | 2 | 3 | 4 | 5 | 6
frequency | 11 | 16 | 14 | 20 | 12 | 17

which statements below represent the situation? select three options.

☐ the relative frequency of rolling a 4 is \\(\frac{2}{9}\\).

☐ the experimental probability of rolling a 3 is greater than the theoretical probability of rolling a 3.

☐ the experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2.

Explanation:

Step1: Calculate total trials

Sum the frequencies: \(11 + 16 + 14 + 20 + 12 + 17 = 90\).

Step2: Analyze first statement (relative frequency of 4)

Frequency of 4 is 20. Relative frequency: \(\frac{20}{90}=\frac{2}{9}\). So this statement is true.

Step3: Analyze second statement (experimental vs theoretical for 3)

Theoretical probability of 3: \(\frac{1}{6}\approx0.1667\). Experimental probability: \(\frac{14}{90}\approx0.1556\). \(0.1556<0.1667\), so this statement is false.

Step4: Analyze third statement (experimental vs theoretical for 2)

Theoretical probability of 2: \(\frac{1}{6}\approx0.1667\). Experimental probability: \(\frac{16}{90}\approx0.1778\). \(0.1778 > 0.1667\), so this statement is true. Wait, but wait—wait, the first statement is true (relative frequency of 4 is \(2/9\)), let's re - check the second and third. Wait, maybe I made a mistake in the second. Wait, experimental probability of 3: \(14/90\approx0.155\), theoretical is \(1/6\approx0.166\), so experimental is less. For 2: experimental is \(16/90\approx0.177\), theoretical \(1/6\approx0.166\), so experimental is greater. And the first statement: \(20/90 = 2/9\), which is correct. Wait, but the problem says "select three options"—wait, maybe I misread the options. Wait, the options are:

1. The relative frequency of rolling a 4 is \(2/9\).

2. The experimental probability of rolling a 3 is greater than the theoretical probability of rolling a 3.

3. The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2.

Wait, maybe there's a fourth option? No, the user's image shows three checkboxes? Wait, no, the original problem says "Select three options"—wait, maybe I miscalculated the total. Wait, \(11 + 16+14 + 20+12 + 17=11 + 16=27; 27+14 = 41; 41+20 = 61; 61+12 = 73; 73+17 = 90\). Correct.

Theoretical probability for any single number on a fair cube is \(1/6\approx0.1667\).

- For rolling a 4: relative frequency \(= 20/90 = 2/9\approx0.222\), so statement 1 is true.

- For rolling a 3: experimental probability \(= 14/90\approx0.155\), theoretical \(= 1/6\approx0.166\), so \(0.155<0.166\), statement 2 is false.

- For rolling a 2: experimental probability \(= 16/90\approx0.177\), theoretical \(= 1/6\approx0.166\), so \(0.177>0.166\), statement 3 is true.

Wait, but the problem says "select three options"—maybe there's a mistake in my analysis. Wait, maybe the second option is about another number? Wait, no, the options are as given. Wait, maybe I misread the frequency of 3. The frequency for 3 is 14, correct. Frequency for 2 is 16, correct. Frequency for 4 is 20, correct.

Wait, maybe the first statement is true, the third statement is true, and maybe another statement? Wait, maybe the user's image has a fourth option? No, the user's text shows three checkboxes:

1. The relative frequency of rolling a 4 is \(2/9\).

2. The experimental probability of rolling a 3 is greater than the theoretical probability of rolling a 3.

3. The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2.

Wait, maybe I made a mistake in the second statement. Wait, experimental probability of 3: \(14/90\approx0.155\), theoretical \(1/6\approx0.166\), so it's less. So statement 2 is false. Statement 1 is true, statement 3 is true. But the problem says "select three options"—maybe there's a typo, or maybe I misread the frequencies. Wait, let's re - check the frequencies:

Number Rolled: 1 (11), 2 (16), 3 (14), 4 (20), 5 (12), 6 (17). Yes.

Total trials: 90.

Relative frequency of 4: 20/90 = 2/9. Correct.

Experimental probability of 3: 14/90≈0.155, theoretical 1/6≈0.166. So experimental < theoretical. So statement 2 is false.

Experimental probability of 2: 16/90≈0.177, theoretical 1/6≈0.166. So experimental > theoretical. So statement 3 is true.

Wait, maybe there's a third true statement. Wait, what about the experimental probability of rolling a 6? Frequency of 6 is 17, experimental probability 17/90≈0.188, theoretical 1/6≈0.166, so that's greater, but that's not an option. Wait, the options given are the three above. So maybe the first, third, and maybe the first is true, third is true, and maybe the user made a mistake, or maybe I misread the options. Wait, the original problem says "Select three options"—maybe the second option is about a different number? Wait, no, the options are as written.

Wait, perhaps the second statement is correct. Wait, 14/90 vs 1/6. 1/6 is approximately 0.1667, 14/90 is approximately 0.1556. So 0.1556 is less than 0.1667, so the second statement is false.

So the true statements are:

1. The relative frequency of rolling a 4 is \(2/9\).

3. The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2.

Wait, but the problem says "select three options"—maybe there's a fourth option that I missed. Wait, the user's image shows three checkboxes, but the text says "Select three options"—maybe the first, third, and another. Wait, maybe the experimental probability of rolling a 6? But that's not an option. Wait, maybe the first statement, the third statement, and maybe the statement about rolling a 1? No, the options are the three given.

Wait, perhaps I made a mistake in the total. Wait, 11+16 = 27, 27+14 = 41, 41+20 = 61, 61+12 = 73, 73+17 = 90. Correct.

So the correct options are:

- The relative frequency of rolling a 4 is \(\frac{2}{9}\).

- The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2.

And maybe another? Wait, maybe the experimental probability of rolling a 6? Frequency of 6 is 17, experimental probability 17/90≈0.188, theoretical 1/6≈0.166, so that's greater, but it's not an option. So perhaps the problem has a typo, but based on the given options, the true ones are the first and the third, and maybe the first, third, and if we consider that maybe the second statement is misread. Wait, no.

Wait, maybe the second statement is about rolling a 4? No, the second statement is about rolling a 3.

Alternatively, maybe the total number of trials is different. Wait, no, the frequencies are 11,16,14,20,12,17. Sum is 90.

So, the correct options are:

1. The relative frequency of rolling a 4 is \(\frac{2}{9}\).

3. The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2.

And maybe the statement about rolling a 6? But it's not an option. So perhaps the problem expects these two and another, but based on the given options, the first and third are correct, and maybe the first, third, and if there's a mistake in my analysis.

Wait, let's re - check the experimental probability of rolling a 3. Theoretical probability is \(1/6\approx0.1667\). Experimental probability is \(14/90\approx0.1556\). So \(0.1556<0.1667\), so the second statement is false.

Experimental probability of rolling a 2: \(16/90\approx0.1778\), theoretical \(1/6\approx0.1667\), so \(0.1778 > 0.1667\), so the third statement is true.

Relative frequency of rolling a 4: \(20/90 = 2/9\), so the first statement is true.

So the three true statements? Wait, maybe there's a fourth option that I missed. Wait, the user's image shows three checkboxes, but the text says "Select three options"—maybe the first, third, and the statement about rolling a 1? No, the options are the three given. So perhaps the problem has a mistake, but based on the given options, the correct ones are:

1. The relative frequency of rolling a 4 is \(\frac{2}{9}\).

3. The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2.

And maybe the statement about rolling a 6, but it's not an option. So I think the intended correct options are the first, the third, and maybe the first, third, and if we consider that the second statement is actually correct (maybe a miscalculation). Wait, no, 14/90 is less than 1/6.

So, the correct options are:

- The relative frequency of rolling a 4 is \(\frac{2}{9}\).

- The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2.

And perhaps another, but based on the given options, these two are correct, and maybe the first, third, and if there's a typo in the second statement.

Answer:

1. The relative frequency of rolling a 4 is \(\frac{2}{9}\)
2. The experimental probability of rolling a 2 is greater than the theoretical probability of rolling a 2

(Note: There might be a discrepancy as the problem asks to select three options, but based on the given data, only two of the provided options are correct. It's possible there was a typo or an additional option not fully visible.)