louise flips a coin and rolls a 6-sided die 36 times. the sample space for her experiment is below. compare the theoretical probability of flipping a head and rolling an even number to the empirical probability of the experiment. what would we expect if we did many more trials?

louises experiment
sample space

h 3 | t 2 | t 3 | t 6 | t 6 | t 3
h 1 | t 2 | h 1 | h 5 | h 4 | h 1
t 4 | h 2 | t 3 | h 5 | h 4 | t 3
t 6 | t 3 | h 4 | t 1 | h 3 | h 2
t 6 | h 2 | h 6 | t 6 | h 1 | h 1
h 1 | h 4 | t 1 | h 2 | t 6 | h 6

- the theoretical probability is about 0.28 whereas the empirical probability is 0.25. the theoretical probability was higher. if we did the experiment many more times, we would expect similar results.
- the theoretical probability is 0.25 whereas the empirical probability is about 0.28. the probabilities for both were close. if we did the experiment many more times, we would expect the results to be even closer.
- the theoretical probability is 0.5 whereas the empirical probability is 0.28. the experimental probability was lower. if we did the experiment many more times, we would expect similar results.
- the theoretical probability is about 0.5 whereas the empirical probability is 0.25. the theoretical probability was higher. if we did the experiment many more times, we would expect the probabilities to be even closer.

Question

louise flips a coin and rolls a 6-sided die 36 times. the sample space for her experiment is below. compare the theoretical probability of flipping a head and rolling an even number to the empirical probability of the experiment. what would we expect if we did many more trials?

louises experiment
sample space

h 3 | t 2 | t 3 | t 6 | t 6 | t 3
h 1 | t 2 | h 1 | h 5 | h 4 | h 1
t 4 | h 2 | t 3 | h 5 | h 4 | t 3
t 6 | t 3 | h 4 | t 1 | h 3 | h 2
t 6 | h 2 | h 6 | t 6 | h 1 | h 1
h 1 | h 4 | t 1 | h 2 | t 6 | h 6

- the theoretical probability is about 0.28 whereas the empirical probability is 0.25. the theoretical probability was higher. if we did the experiment many more times, we would expect similar results.
- the theoretical probability is 0.25 whereas the empirical probability is about 0.28. the probabilities for both were close. if we did the experiment many more times, we would expect the results to be even closer.
- the theoretical probability is 0.5 whereas the empirical probability is 0.28. the experimental probability was lower. if we did the experiment many more times, we would expect similar results.
- the theoretical probability is about 0.5 whereas the empirical probability is 0.25. the theoretical probability was higher. if we did the experiment many more times, we would expect the probabilities to be even closer.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Empirical Probability" ], "new_concepts": [ "Theoretical Probability", "Law of Large Numbers" ], "current_concepts": [ "Empirical Probability", "Theoretical Probability", "Law of Large Numbers" ] } ## Calculate theoretical probability $$ \begin{aligned} P(\text{Head}) &= 0.5 \ P(\text{Even}) &= \frac{3}{6} = 0.5 \ P(\text{Head and Even}) &= 0.5 \times 0.5 = 0.25 \end{aligned} $$ ## Calculate empirical probability from the 36 trials $$ \begin{aligned} \text{Favorable outcomes (H2, H4, H6):} &\quad \text{H4, H4, H2, H4, H6, H2, H2, H6, H4, H2} \ \text{Count of favorable outcomes:} &\quad 10 \ P_{\text{empirical}} &= \frac{10}{36} \approx 0.28 \end{aligned} $$ ## Compare probabilities and apply the Law of Large Numbers $$ \begin{aligned} P_{\text{theoretical}} &= 0.25 \ P_{\text{empirical}} &\approx 0.28 \ \text{With more trials:} &\quad \text{empirical probability approaches theoretical probability} \end{aligned} $$

(A) The theoretical probability is about 0.28 whereas the empirical probability is 0.25. The theoretical probability was higher. If we did the experiment many more times, we would expect similar results.

(B) The theoretical probability is 0.25 whereas the empirical probability is about 0.28. The probabilities for both were close. If we did the experiment many more times, we would expect the results to be even closer.

(C) The theoretical probability is 0.5 whereas the empirical probability is 0.28. The experimental probability was lower. If we did the experiment many more times, we would expect similar results.

(D) The theoretical probability is about 0.5 whereas the empirical probability is 0.25. The theoretical probability was higher. If we did the experiment many more times, we would expect the probabilities to be even closer.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Theoretical and Empirical Probability" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate theoretical probability

Calculate empirical probability from the 36 trials

Compare probabilities and apply the Law of Large Numbers

Answer: