Question
In a survey, 300 people were asked to choose one card out of five cards labeled 1 to 5. The results are shown in the table. Compare the theoretical probability and experimental probability of choosing a card with the number 3.
Cards Chosen
Number
1
2
3
4
5
Bold Frequency
84
48
96
24
48
Question content area bottom
Part 1
The theoretical probability of choosing a card with the number 3 is
enter your response here%. The experimental probability of choosing a card with the number 3 is
enter your response here%. The theoretical probability is
▼
greater than
less than
equal to
the experimental probability.
(Type integers or decimals.)
Cards Chosen
Number
1
2
3
4
5
Bold Frequency
84
48
96
24
48
Question content area bottom
Part 1
The theoretical probability of choosing a card with the number 3 is
enter your response here%. The experimental probability of choosing a card with the number 3 is
enter your response here%. The theoretical probability is
▼
greater than
less than
equal to
the experimental probability.
(Type integers or decimals.)
Answers
GPT-4o mini
To solve this problem, we need to calculate both the theoretical probability and the experimental probability of choosing a card with the number 3.
**Theoretical Probability:**
The theoretical probability of an event is calculated by taking the number of favorable outcomes divided by the total number of possible outcomes. In this case, since there are 5 cards and only one of them is labeled with the number 3, the theoretical probability is:
\[
P(\text{choosing 3}) = \frac{\text{Number of favorable outcomes (3)}}{\text{Total possible outcomes}}
\]
\[
P(\text{choosing 3}) = \frac{1}{5} = 0.2
\]
To express this as a percentage:
\[
0.2 \times 100 = 20\%
\]
So, the theoretical probability of choosing a card with the number 3 is **20%**.
**Experimental Probability:**
The experimental probability is calculated by taking the number of times the event occurs (in this case, choosing card number 3) over the total number of trials. From the data given, the frequency of choosing card number 3 is 96, and the total number of respondents is 300. Thus, the experimental probability is:
\[
P(\text{choosing 3}) = \frac{\text{Number of times 3 was chosen}}{\text{Total number of respondents}}
\]
\[
P(\text{choosing 3}) = \frac{96}{300}
\]
Calculating this gives:
\[
P(\text{choosing 3}) = 0.32
\]
To express this as a percentage:
\[
0.32 \times 100 = 32\%
\]
So, the experimental probability of choosing a card with the number 3 is **32%**.
**Comparison:**
Now we can summarize the findings:
- Theoretical probability: **20%**
- Experimental probability: **32%**
We can conclude that the theoretical probability is **less than** the experimental probability.
### Summary:
- Theoretical Probability of choosing a card with number 3: **20%**
- Experimental Probability of choosing a card with number 3: **32%**
- The theoretical probability is **less than** the experimental probability.
**Theoretical Probability:**
The theoretical probability of an event is calculated by taking the number of favorable outcomes divided by the total number of possible outcomes. In this case, since there are 5 cards and only one of them is labeled with the number 3, the theoretical probability is:
\[
P(\text{choosing 3}) = \frac{\text{Number of favorable outcomes (3)}}{\text{Total possible outcomes}}
\]
\[
P(\text{choosing 3}) = \frac{1}{5} = 0.2
\]
To express this as a percentage:
\[
0.2 \times 100 = 20\%
\]
So, the theoretical probability of choosing a card with the number 3 is **20%**.
**Experimental Probability:**
The experimental probability is calculated by taking the number of times the event occurs (in this case, choosing card number 3) over the total number of trials. From the data given, the frequency of choosing card number 3 is 96, and the total number of respondents is 300. Thus, the experimental probability is:
\[
P(\text{choosing 3}) = \frac{\text{Number of times 3 was chosen}}{\text{Total number of respondents}}
\]
\[
P(\text{choosing 3}) = \frac{96}{300}
\]
Calculating this gives:
\[
P(\text{choosing 3}) = 0.32
\]
To express this as a percentage:
\[
0.32 \times 100 = 32\%
\]
So, the experimental probability of choosing a card with the number 3 is **32%**.
**Comparison:**
Now we can summarize the findings:
- Theoretical probability: **20%**
- Experimental probability: **32%**
We can conclude that the theoretical probability is **less than** the experimental probability.
### Summary:
- Theoretical Probability of choosing a card with number 3: **20%**
- Experimental Probability of choosing a card with number 3: **32%**
- The theoretical probability is **less than** the experimental probability.