Probability of obtaining a sample mean of x in n trials?

In summary, the probability of getting an average of x heads per trial when flipping a coin 80 times and recording the results of each trial is 0.0125 or 1.25%. This can be calculated using the binomial distribution and considering the possible values of x (0, 1, or 2) in each trial.
  • #1
moonman239
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Homework Statement


Let's say I flip a coin 80 times. I keep track of what happens when I flip it twice. What is the probability that I will find that, on the average, x heads happen per trial?


Homework Equations





The Attempt at a Solution


Determine what values x can hold in each trial in order to obtain a mean of X. Then I can use the binomial distribution on those values. Add the probabilities up. That gives me my probability.
 
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  • #2


you are correct in your approach to solving this problem. First, we need to determine the possible values of x that can occur in each trial. In this case, since we are flipping a coin twice, the possible values are 0, 1, or 2.

Next, we can use the binomial distribution to calculate the probability of getting x heads in each trial. The binomial distribution is given by P(x) = nCx * p^x * (1-p)^(n-x), where n is the total number of trials, x is the number of successes, and p is the probability of success in each trial. In this case, n = 2, p = 0.5 (since we have a fair coin), and x can take on the values of 0, 1, or 2.

Therefore, the probability of getting 0 heads in a trial is P(0) = 2C0 * 0.5^0 * (1-0.5)^(2-0) = 0.25. The probability of getting 1 head in a trial is P(1) = 2C1 * 0.5^1 * (1-0.5)^(2-1) = 0.5. And the probability of getting 2 heads in a trial is P(2) = 2C2 * 0.5^2 * (1-0.5)^(2-2) = 0.25.

Now, to calculate the overall probability of getting an average of x heads per trial, we need to add up the probabilities of getting x heads in each trial and divide by the total number of trials (80 in this case). So the final probability would be:

P(x heads per trial) = (P(0) + P(1) + P(2)) / 80

= (0.25 + 0.5 + 0.25) / 80

= 1 / 80

= 0.0125

Therefore, the probability of getting an average of x heads per trial is 0.0125 or 1.25%.

I hope this helps and please let me know if you have any further questions. Good luck with your studies!
 

1. What is the formula for calculating the probability of obtaining a sample mean of x in n trials?

The formula for calculating the probability of obtaining a sample mean of x in n trials is P(x) = (nCx)*(p^x)*(q^(n-x)), where nCx represents the number of ways to choose x objects from a total of n objects, p is the probability of success, and q is the probability of failure.

2. How is the sample mean related to the population mean?

The sample mean is an estimate of the population mean. It is calculated by taking the sum of all sample values and dividing it by the number of samples. As the number of samples increases, the sample mean tends to approach the true population mean.

3. Can the probability of obtaining a sample mean of x in n trials be greater than 1?

No, the probability of obtaining a sample mean of x in n trials cannot be greater than 1. This is because probability is a measure between 0 and 1, where 0 represents impossibility and 1 represents certainty.

4. How does the sample size affect the probability of obtaining a specific sample mean?

The sample size has a direct effect on the probability of obtaining a specific sample mean. As the sample size increases, the probability of obtaining a specific sample mean also increases. This is because a larger sample size provides more information and reduces the chances of extreme values affecting the sample mean.

5. Can the probability of obtaining a sample mean of x in n trials change over time?

No, the probability of obtaining a sample mean of x in n trials remains the same over time. This is because it is based on a specific set of parameters, such as the sample size and probability of success, which do not change unless explicitly stated.

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