A local church lottery states that the probability that any ticket wins a prize is 36%. Suppose that you are generous and purchase 1225 tickets, what is the probability that less than 441 of the tickets result in A prize? Give your answer as a percentage.
A) 25%
B) 34%
C) 68%
D) 95%
E) None of the above.

Question

Quizplus · Accepted Answer

Given the large number of tickets (1225) and a specific probability (36%), the scenario can be approximated using the normal distribution due to the Central Limit Theorem. The mean (μ) of the distribution of winning tickets is given by n*p, where n is the number of trials (1225 tickets) and p is the probability of success (0.36). Thus, μ = 1225 * 0.36 = 441. The standard deviation (σ) is given by sqrt(n*p*(1-p)), which calculates to sqrt(1225*0.36*0.64) ≈ 18.3.To find the probability of winning less than 441 tickets, we look for the probability of getting results below the mean in a normal distribution, which is essentially asking for the probability of a result less than or equal to the mean. In a perfectly symmetrical distribution, this would be 50%. However, the question asks for the probability of less than 441, not less than or equal to. In practical terms, for a large sample like this, the difference between less than 441 and less than or equal to 441 is negligible. However, the options provided do not account for this specific scenario accurately, as they seem to suggest a specific calculation or approximation that doesn't directly apply to the information given. The correct approach using normal approximation would involve calculating a z-score and then looking up that value in a standard normal distribution table or using a calculator. Since the exact calculation or approximation isn't provided in the options, and given the nature of the question, the most accurate answer with the information provided is "None of the above."

A Local Church Lottery States That the Probability That Any