To find the theoretical and experimental probabilities of choosing a card with the number 3, follow these steps:
Part 1: Theoretical Probability
The theoretical probability of choosing any specific card (like card number 3) among five cards (1 through 5) is calculated as follows:
\[ \text{Theoretical Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]
Here, there is 1 favorable outcome (choosing card number 3) out of 5 total outcomes:
\[ \text{Theoretical Probability} = \frac{1}{5} = 0.2 \text{ or } 20% \]
Part 2: Experimental Probability
The experimental probability is calculated based on the results of the survey. The number of times card number 3 was chosen is given as 48. The total number of surveys (people asked) was 200. Therefore:
\[ \text{Experimental Probability} = \frac{\text{Number of times card 3 was chosen}}{\text{Total number of people surveyed}} = \frac{48}{200} \]
Calculating this gives:
\[ \text{Experimental Probability} = 0.24 \text{ or } 24% \]
Summary of Results
- Theoretical Probability of choosing card number 3: 20%
- Experimental Probability of choosing card number 3: 24%
Comparison
Now, to compare the two probabilities:
- Theoretical Probability is 20%.
- Experimental Probability is 24%.
Since \( 20% < 24% \), we conclude that:
The theoretical probability is less than the experimental probability.
Final Answers:
- The theoretical probability of choosing a card with the number 3 is 20%.
- The experimental probability of choosing a card with the number 3 is 24%.
- The theoretical probability is less than the experimental probability.