Find the exact formula for P(Sn=k)

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In summary, the conversation discusses finding the exact formula for the probability of obtaining a sum of k after n draws at random with replacement from a box containing four tickets with numbers 0, 1, 1, and 2. The conversation mentions using the trinomial theorem to simplify the formula for P(Sn=k).
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A box contains four tickets, numbered 0,1,1, and 2. Let Sn be the sum of the numbers obtained from n draws at random with replacement from the box.
Find the exact formula for P(Sn=k) (k=0,1,2...)

I started by finding the probability of the first few sums:
P(Sn=0) = (1/4)^n
P(Sn=1) = n * (1/4)^(n-1) * (1/2)
P(Sn=2) = n * (1/4)^(n-1)*(1/4) + (n choose 2) * (1/4)^(n-2)(1/2)^2

Then I get stuck, since the combinations of draws that lead to the subsequent draws get much more complicated.
Is there a more simplified way to find the formula for P(Sn=k)?
 
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1. What does P(Sn=k) represent?

P(Sn=k) represents the probability of getting exactly k successes in n trials.

2. How do you find the exact formula for P(Sn=k)?

The exact formula for P(Sn=k) is given by the binomial distribution formula: P(Sn=k) = nCk * p^k * (1-p)^(n-k), where n is the number of trials, p is the probability of success, and nCk is the combination formula (n choose k).

3. Can the formula for P(Sn=k) be used for any type of experiment or scenario?

No, the formula for P(Sn=k) can only be used for binomial experiments, which have the following characteristics: fixed number of trials, independent trials, two possible outcomes (success or failure), and constant probability of success for each trial.

4. How can the formula for P(Sn=k) be applied in real-life situations?

The formula for P(Sn=k) can be used to calculate the probability of getting a certain number of successes in a given number of trials. For example, it can be used in quality control to determine the probability of a certain number of defective products in a batch, or in medicine to calculate the probability of a certain number of successful treatments in a sample of patients.

5. Are there any limitations to the formula for P(Sn=k)?

Yes, the formula for P(Sn=k) assumes that the trials are independent and that the probability of success remains constant for each trial. In real-life situations, these assumptions may not always hold, which can affect the accuracy of the calculated probability.

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