PLEASE HELP ME :(
Brandon flips two fair coins multiple times and records the frequency of the results in the following table.
50 trials|125trials|150trials
HH 10 33 37
TH 9 31 38
HT 13 28 38
TT 18 33 37
DETERMINE THE EXPERIMENTAL PROBABILITY AND THEORETICAL PROBABILITY FOR EACH OUTCOME IN EACH SET OF BRANDON'S TRIALS. HOW DO THE EXPERIMANTAL PRBABILITY AND THEORECTICAL PROBABILITY RELATE TO EACH OTHER FOR EACH OF THE 3 SETS OF TRIALS?
Explain your answer, providing calculations or any other strategy that will support your reasoning.
To determine the experimental and theoretical probabilities for each outcome in each set of trials, we need to use the given data.
Let's start by calculating the experimental probability of each outcome. Experimental probability is determined by dividing the frequency of a specific outcome by the total number of trials. In this case, the total number of trials is given in the table.
For the first set of trials (50 trials):
- Experimental probability of HH: 10/50 = 0.2
- Experimental probability of TH: 9/50 = 0.18
- Experimental probability of HT: 13/50 = 0.26
- Experimental probability of TT: 18/50 = 0.36
For the second set of trials (125 trials):
- Experimental probability of HH: 33/125 = 0.264
- Experimental probability of TH: 31/125 = 0.248
- Experimental probability of HT: 28/125 = 0.224
- Experimental probability of TT: 33/125 = 0.264
For the third set of trials (150 trials):
- Experimental probability of HH: 37/150 = 0.24666...
- Experimental probability of TH: 38/150 = 0.25333...
- Experimental probability of HT: 38/150 = 0.25333...
- Experimental probability of TT: 37/150 = 0.24666...
Now, let's calculate the theoretical probability. The theoretical probability of an outcome is determined by the ratio of the favorable outcomes to the total possible outcomes, assuming an unbiased fair coin. Since there are two coins being flipped, each coin has two possible outcomes (heads or tails). Therefore, there are 2 * 2 = 4 possible outcomes for each trial.
For all sets of trials, the theoretical probability for each outcome is 1/4 = 0.25.
Now, let's analyze the relationship between the experimental probability and the theoretical probability for each set of trials:
For the first set of trials, we can see that the experimental probabilities for each outcome differ slightly from the theoretical probability of 0.25. This is expected due to the small number of trials (50 trials), which may not be sufficient to converge to the true probabilities.
For the second set of trials, the experimental probabilities are closer to the theoretical probability. The larger number of trials (125 trials) allows for more accurate estimation of the probabilities.
For the third set of trials, the experimental probabilities are even closer to the theoretical probability, suggesting a stronger convergence as the number of trials increases (150 trials).
In general, as the number of trials increases, the experimental probabilities tend to converge to the theoretical probabilities. This is known as the Law of Large Numbers, which states that as the number of trials increases, the experimental probabilities approach the theoretical probabilities.
To determine the experimental probability and theoretical probability for each outcome in each set of Brandon's trials, we can use the given table.
Let's start with the first set of trials: 50 trials.
Experimental Probability:
To find the experimental probability, we divide the frequency of each outcome by the total number of trials.
For example:
- HH occurred 10 times out of 50 trials,
so the experimental probability of HH is 10/50 = 0.2.
Calculating in the same way for the remaining outcomes, we get the following experimental probabilities for the first set of trials:
HH: 0.2
TH: 0.18
HT: 0.26
TT: 0.36
Theoretical Probability:
For two fair coins, there are a total of 2^2 = 4 possible outcomes (HH, TH, HT, TT), each with an equal probability of 1/4.
So, the theoretical probability for each outcome in the first set of trials is:
HH: 1/4 = 0.25
TH: 1/4 = 0.25
HT: 1/4 = 0.25
TT: 1/4 = 0.25
Now let's move to the second set of trials: 125 trials.
Using the same process as before, we can find the experimental and theoretical probabilities for each outcome in the second set of trials.
Experimental Probability:
HH: 33/125 = 0.264
TH: 31/125 = 0.248
HT: 28/125 ≈ 0.224
TT: 33/125 = 0.264
Theoretical Probability:
HH: 0.25
TH: 0.25
HT: 0.25
TT: 0.25
Finally, let's move to the third set of trials: 150 trials.
Again, following the same process as before, we can find the experimental and theoretical probabilities for each outcome in the third set of trials.
Experimental Probability:
HH: 37/150 ≈ 0.2467
TH: 38/150 ≈ 0.2533
HT: 38/150 ≈ 0.2533
TT: 37/150 ≈ 0.2467
Theoretical Probability:
HH: 0.25
TH: 0.25
HT: 0.25
TT: 0.25
Now, let's analyze the relation between experimental probability and theoretical probability for each of the 3 sets of trials.
In all three sets, we observe that the theoretical probability for each outcome is the same (0.25), since we are dealing with fair coins.
On the other hand, the experimental probabilities, which are based on the frequencies observed in each set of trials, vary slightly. However, they tend to approach the theoretical probabilities with an increasing number of trials.
This is expected since as the number of trials increases, the experimental probabilities should become closer to the theoretical probabilities.
In summary, the experimental and theoretical probabilities relate to each other by showing that the experimental probabilities tend to approach the theoretical probabilities as the number of trials increases.