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Conducting a Simulation of a Binomial Probability Distribution




$X$		012345

$P(X)$		000.20.30.5






Spin the Wheel 5 Times




Number of Trials

10







Trial		Results

1		5,2,2,5,9

2		0,1,4,9,3

3		8,3,5,6,4

4		5,2,1,8,4

5		1,2,2,7,8

6		1,8,6,6,5

7		9,4,1,5,3

8		1,7,7,2,8

9		8,3,8,1,3

10		3,4,2,0,6






Calculate how many times $X=5$ is presented in the column "How many like basketball?" and divide by 10.

Verified Answer :

  • Count the number of times 5 appears in the 10 trials, which is \(6\).
  • Divide the total count by the number of trials, which is \(10\).
  • Calculate the experimental probability as \(\frac{6}{10}\).
  • The final experimental probability is \(\boxed{0.6}\).

Explanation

  1. Analyze the data and count occurrences of 5

To determine the experimental frequency of the value \(X=5\) based on the provided simulation, we first analyze the results table given for the \(10\) trials. Each trial consists of \(5\) generated numbers. We need to count every occurrence of the number \(5\) across all these trials.

  1. Calculate the total frequency of 5

Let's examine the results for each trial:

  • Trial 1: 5, 2, 2, 5, 9 (contains two 5s)
  • Trial 2: 0, 1, 4, 9, 3 (contains zero 5s)
  • Trial 3: 8, 3, 5, 6, 4 (contains one 5)
  • Trial 4: 5, 2, 1, 8, 4 (contains one 5)
  • Trial 5: 1, 2, 2, 7, 8 (contains zero 5s)
  • Trial 6: 1, 8, 6, 6, 5 (contains one 5)
  • Trial 7: 9, 4, 1, 5, 3 (contains one 5)
  • Trial 8: 1, 7, 7, 2, 8 (contains zero 5s)
  • Trial 9: 8, 3, 8, 1, 3 (contains zero 5s)
  • Trial 10: 3, 4, 2, 0, 6 (contains zero 5s)

Summing these up, the total number of times the value \(5\) appears in the results is \(2 + 0 + 1 + 1 + 0 + 1 + 1 + 0 + 0 + 0 = 6\).

  1. Calculate the final probability

Now that we have the total frequency of the value \(5\), which is \(N_5 = 6\), we calculate the probability by dividing this frequency by the total number of trials conducted, which is \(10\).

The experimental probability is:
\[\frac{N_5}{10} = \frac{6}{10} = 0.6\]
Thus, the experimental probability of obtaining the value \(5\) in this simulation is \(0.6\).

Examples

This type of simulation is commonly used to test theoretical probability distributions against real-world observations. For example, if you are analyzing a game of chance or a quality control process, you can compare your theoretical model (the probability distribution) with the actual frequency of outcomes (the simulation data) to see if the system is behaving as expected. By calculating \(\frac{\text{observed frequency}}{\text{total trials}}\), you can validate your model or identify biases in your data-generating process.

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