Explain the variation between the experimental and theoretical relative frequencies.
The experimental relative frequency is the proportion of times an event or outcome occurs in a repeated experiment, while the theoretical relative frequency is the expected proportion of times the same event or outcome should occur based on probability theory. The variation between the experimental and theoretical relative frequencies arises due to sampling variability, measurement errors, and other factors that affect the actual outcomes in the experiment. In some cases, the experimental relative frequency may be close to the theoretical relative frequency if the sample size is large and the experiment is designed correctly. However, in many cases, there may be significant deviations from the theoretical predictions due to random factors and other sources of error. The variation between the experimental and theoretical relative frequencies can be analyzed and quantified using statistical methods such as hypothesis testing, confidence intervals, and regression analysis.
Conduct the experiment flipping two coins 10 times and record the results in the frequency table below. Outcomes are sample spaces identified in task (b).
Outcomes Tally Frequency
HH IIIII/II 7
TT IIIII/ 5
HT II 2
TH IIIII/I 6
Total 20 20
Calculate the relative frequency for each outcome based on your results from task (e).
The relative frequency to HH is 7/20 The relative frequency of HT is 2/20 and the frequency for TH is 6/20 But TT had 5/20 witch is the theoretical frequency of HH,TH,HT,TT.
Compare the theoretical probability with the experimental relative frequency for flipping two tails. Are they the same?
No, they are not the same. The theoretical probability of flipping two tails in two coin flips is 1/4 or 0.25, which means that in a large number of trials, we would expect two tails to occur 25% of the time. However, in this particular experiment where the coins were flipped only 10 times, the experimental relative frequency of flipping two tails is 5/20 or 0.25, which means that two tails were obtained in 25% of the trials. Therefore, in this case, the experimental relative frequency is equal to the theoretical probability for flipping two tails.
Conduct the experiment flipping two coins 20 times and record the results in the frequency table below. Outcomes are sample spaces identified in task (b).
Outcomes Tally Frequency
HH IIIII/II 7
TT IIIII/ 5
HT II 2
TH IIIII/I 6
Total 20 20
Calculate the relative frequency for each outcome based on your results from task (e).
The relative frequency to HH is 7/20 The relative frequency of HT is 2/20 and the frequency for TH is 6/20 But TT had 5/20 witch is the theoretical frequency of HH,TH,HT,TT.
Compare the theoretical probability with the experimental relative frequency for flipping two tails. Are they the same?
No, they are not the same. The theoretical probability of flipping two tails in two coin flips is 1/4 or 0.25, which means that in a large number of trials, we would expect two tails to occur 25% of the time. However, in this particular experiment where the coins were flipped 20 times, the experimental relative frequency of flipping two tails is not provided in the table.
The variation between experimental and theoretical relative frequencies refers to the difference observed when comparing the expected outcomes of an experiment (the theoretical relative frequencies) with the actual outcomes obtained through experimentation (the experimental relative frequencies).
To understand this variation, let's first understand what these terms mean:
- Experimental Relative Frequency: This is the proportion of times an event occurs in an experiment, based on the actual data collected during the experiment.
- Theoretical Relative Frequency: This is the expected proportion of times an event should occur in an experiment, based on mathematical calculations, assumptions, or theoretical models.
Now, let's explore how to calculate both the experimental and theoretical relative frequencies:
1. Experimental Relative Frequency:
- Conduct an experiment or collect data on a particular event or phenomenon multiple times.
- Count the number of times the event of interest occurs.
- Divide the number of occurrences of the event of interest by the total number of trials conducted.
- The result is the experimental relative frequency.
2. Theoretical Relative Frequency:
- In theoretical models or using mathematical calculations, determine the probabilities or expected frequencies of different outcomes.
- Add up the probabilities or expected frequencies of the specific event of interest.
- The result is the theoretical relative frequency.
Now, after calculating both the experimental and theoretical relative frequencies, you can compare and analyze the variation between them. This variation can arise due to several factors:
1. Sampling Variability: If the experiment has a limited number of trials or data points, the experimental relative frequency may not precisely match the theoretical relative frequency. This is because of chance fluctuations or random errors in the sample selection.
2. Systematic Bias: If there are errors or biases in the experimental setup or data collection process, the observed frequencies may deviate from the expected values. This can lead to variations between experimental and theoretical relative frequencies.
3. Incomplete Model: The theoretical model used to calculate the expected frequencies might not fully capture all the complexities and nuances of the real-world experiment. As a result, the theoretical relative frequency may not precisely match the observed experimental relative frequency.
Analyzing the variation between experimental and theoretical relative frequencies can provide insights into the accuracy and reliability of the theoretical model used, the quality of the data collected during the experiment, and help identify any potential biases or errors in the experimental process.