Solution Found!
Consider the birthdays of the students in a class of size
Chapter 1, Problem 11E(choose chapter or problem)
Consider the birthdays of the students in a class of size r. Assume that the year consists of 365 days.
(a) How many different ordered samples of birthdays are possible (r in sample) allowing repetitions (with replacement)?
(b) The same as part (a), except requiring that all the students have different birthdays (without replacement)?
(c) If we can assume that each ordered outcome in part (a) has the same probability, what is the probability that at least two students have the same birthday?
(d) For what value of r is the probability in part (c) about equal to 1/2? Is this number surprisingly small? Hint: Use a calculator or computer to find r.
Questions & Answers
QUESTION:
Consider the birthdays of the students in a class of size r. Assume that the year consists of 365 days.
(a) How many different ordered samples of birthdays are possible (r in sample) allowing repetitions (with replacement)?
(b) The same as part (a), except requiring that all the students have different birthdays (without replacement)?
(c) If we can assume that each ordered outcome in part (a) has the same probability, what is the probability that at least two students have the same birthday?
(d) For what value of r is the probability in part (c) about equal to 1/2? Is this number surprisingly small? Hint: Use a calculator or computer to find r.
ANSWER:Step 1 of 4
We consider the birthdays of the students in a class of size r.
We assume that the year consists of 365 days.
a). Now we have to find how many different ordered samples are possible (r in sample ) allowing repetitions.
Each of r students may have birthday on any of 365 days.
Therefore the number of samples birthdays of size r with replacement is
Number of different ordered sample = \(365 \times 365 \times 365 \times\)...r times.
Number of different ordered sample = \(365^{r}\)
Therefore \(365^{r}\) different ordered samples are possible allowing repetitions.