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Birthday problem). Assume that a host invites n guests and that days of all n+1 people...
The birthday problem is as follows: given a group of n people in a room, what...
The birthday problem is as follows: given a group of n people in a room, what is the probability that two or more of them have the same birthday? It is possible to determine the answer to this question via simulation. Using the starting template provided to you, complete the function called calc birthday probability that takes as input n and returns the probability that two or more of the n people will have the same birthday. To do this,...
(The Birthday Problem) It is known that there are as many as n people in a...
(The Birthday Problem) It is known that there are as many as n people in a room a. How big is the chance that no one has the same birthday? b. What is the value of n so that the probability of problem (a) is less than 50%?
Suppose there are n students in a class. Assume n ≤ 365, and all possible sequences...
Suppose there are n students in a class. Assume n ≤ 365, and all possible sequences of n birthdays are equally likely. We also assume that the birthday of a student is equally likely to be any one of the 365 days of the year (i.e., ignore leap years or seasonal variation in birth rate). (a) What is the probability that all n students have different birthdays? (b) Calculate this probability numerically when n = 23, 45, 65 and 70....
in python The birthday paradox says that the probability that two people in a room will...
in python The birthday paradox says that the probability that two people in a room will have the same birthday is more than half, provided n, the number of people in the room, is more than 23. This property is not really a paradox, but many people find it surprising. Design a Python program that can test this paradox by a series of experiments on randomly generated birthdays, which test this paradox for n = 5, 6, 7, ..., 50....
4. For the following problem, ignore the year of birth while comparing two birthdays. Moreover, assume...
4. For the following problem, ignore the year of birth while comparing two birthdays. Moreover, assume that the year is exactly 365 days (ignore the 29th of February) Note that matching birthdays means the birthdays are the same (a) You are a member of a class room that has n +1 students including you. What is the probability that you find at least one student, other than you, who has a birthday that matches yours? (b) In another classroom that...
4. For the following problem, ignore the year of birth while comparing two birthdays. Moreover, assume...
4. For the following problem, ignore the year of birth while comparing two birthdays. Moreover, assume that the year is exactly 365 days (ignore the 29th of February) Note that matching birthdays means the birthdays are the same (a) You are a member of a class room that has n 1 students including you. What is the probability that you find at least one student, other than you, who has a birthday that matches yours? (b) In another classroom that...
1. The birthday of six random people has been checked. Find the probability that (a) At...
1. The birthday of six random people has been checked. Find the probability that (a) At least one of them is born in September. (b) All five are born in the Spring. Spring here means one of the month March, April, or May. (c) At least two of them are born in the same month. In this problem you can assume that a year is 365 days. 2.A fair die is rolled three times. We say that a match has...
Problem 1. Suppose that we take a random sequence of numbers modulo n. The proba- bility...
Problem 1. Suppose that we take a random sequence of numbers modulo n. The proba- bility that the first k terms are all distinct is n - 1 п 2 n - (k – 1) P(n, k) = 2 п п п as we saw in class. For this problem, you will want to write a (very short) program to compute this number, but you do not need to submit it this time. (For both parts, you can submit a...
of 6 23. The pregnancies of pigs are normally distributed with a mean of 110 days...
of 6 23. The pregnancies of pigs are normally distributed with a mean of 110 days and a standard deviation of 11 days. What is the probability that a pregnancy lasts less than 108 days? a. Sketch the graph b. Find the probability 24. A study of the amount of time it takes a to cook a steak shows that the mean is 7 minutes and the standard deviation is 2 minutes. If 14 steaks are randomly selected, find the...
Problem 5 5.a Consider the following identity. For all positive integers n and k with n...
Problem 5 5.a Consider the following identity. For all positive integers n and k with n 2k, (n choose k) + (n choose k-1) = (n+1 choose k). This can be demonstrated either algebraically or via a story proof. To prove the identity algebraically, we can write (n choose k) + (n choose k-1) = n!/[k!(n-k)!] + n!/[(k-1)!(n-k+1)!] = [(n-k+1)n! + (k)n!]/[k!(n-k+1)!] [n!(n+1)/k!(n-k+1)!] = (n+1 choose k). Which of the following is a story proof of the identity? Consider a...
4. For the following problem, ignore the year of birth while comparing two birthdays. Moreover, assume that the year is exactly 365 days (ignore the 29th of February) Note that matching birthdays means the birthdays are the same (a) You are a member of a class room that has n +1 students including you. What is the probability that you find at least one student, other than you, who has a birthday that matches yours? (b) In another classroom that...
4. For the following problem, ignore the year of birth while comparing two birthdays. Moreover, assume that the year is exactly 365 days (ignore the 29th of February) Note that matching birthdays means the birthdays are the same (a) You are a member of a class room that has n 1 students including you. What is the probability that you find at least one student, other than you, who has a birthday that matches yours? (b) In another classroom that...
Problem 1. Suppose that we take a random sequence of numbers modulo n. The proba- bility that the first k terms are all distinct is n - 1 п 2 n - (k – 1) P(n, k) = 2 п п п as we saw in class. For this problem, you will want to write a (very short) program to compute this number, but you do not need to submit it this time. (For both parts, you can submit a...
of 6 23. The pregnancies of pigs are normally distributed with a mean of 110 days and a standard deviation of 11 days. What is the probability that a pregnancy lasts less than 108 days? a. Sketch the graph b. Find the probability 24. A study of the amount of time it takes a to cook a steak shows that the mean is 7 minutes and the standard deviation is 2 minutes. If 14 steaks are randomly selected, find the...
Problem 5 5.a Consider the following identity. For all positive integers n and k with n 2k, (n choose k) + (n choose k-1) = (n+1 choose k). This can be demonstrated either algebraically or via a story proof. To prove the identity algebraically, we can write (n choose k) + (n choose k-1) = n!/[k!(n-k)!] + n!/[(k-1)!(n-k+1)!] = [(n-k+1)n! + (k)n!]/[k!(n-k+1)!] [n!(n+1)/k!(n-k+1)!] = (n+1 choose k). Which of the following is a story proof of the identity? Consider a...
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