1. Simulate 20 different groups of 50 birthdays using technology. 2. Use the res
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Question
1. Simulate 20 different groups of 50 birthdays using technology.
2. Use the results to estimate the probability that among 50 randomly selected people, at least 3 have the same birthday.
Project 2 (from the textbook) Classic Birthday Problem Describe the purpose of the study. Explain why you are interested in this simulation Describe Classic Birthday Problem Use technology to simulate 20 different groups of 50 birthdays Use the results to estimate the probability that among 50 randomly selected people, at least 3 have the same birthday Write the statement of conclusions. Determine whether the results have statistical significance. Determine whether the results have practical significance. Applying Simulation Methods Use the simulation method explained below, to randomly generate 20 different groups of 50 birthdays. Use the results to estimate the probability that among 50 randomly selected people, at least 3 have the same birthday. Simulations Calculating probabilities can often be painfully difficult, but simulations provide us with a very practical alternative to calculations based on formal rules. A simulation of a procedure is a process that behaves the same way as the procedure so that similar results are produced. Instead of calculating the probability of getting exactly 5 boys in 10 births, you could repeated- y toss 10 coins and count the number of times that 5 heads (or simulated "boys") occur. Better yet, you could do the simulation with a random number generator on a computer or calculator to randomly generate 1s (or, simulated "boys") and 0s (or simulated "girls"). Let's consider this probability exercise: Classic Birthday Problem Find the probability that among 25 randomly selected people, at least 3 have the same birthday. For the above classic birthday problem, a simulation begins by representing birthdays by integers from 1 through 365, where 1 represents a birthday of January 1, and 2 represents January 2, and so on. We can simulate 50 birthdays by using a calculator or com puter to generate 50 random numbers (with repetition allowed) between 1 and 365. Those numbers can then be sorted, so it becomes easy to examine the list to determine whether any 3 of the simulated birth dates are the same. (After sorting, equal numbers are adjacent.) We can repeat the process as many times as we like, until we are satisfied that we have a good estimate of the probability. There are several ways of obtaining randomly generated numbers from 1 through 365.Explanation / Answer
Let us consider there are 365 days.
probability of 1 people have any birth day = 365/365
probability of 2 people have different birthday = (365/365) * (364/365)
probability of 3 people have different birthday = (365/365)*(364/365)*(363/365)
= 0.991
Probability of 3 people have same birthday = 1- probability of different birthday
= 1- 0.991
= 0.008
Probablity of selecting the 3 people from 25 people = 3/25 = 0.12
=> Probability of randomly selecting 25 people, out of which 3 people have same birthday = 0.12 * 0.008
= 0.00096
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