Consider a group of n people. (a) Explain why the following...

Consider a group of $n$ people. (a) Explain why the following pattern gives the probability that the $n$ people have distinct birthdays. $$\begin{array}{l} n=2: \frac{365}{365} \cdot \frac{364}{365}=\frac{365 \cdot 364}{365^{2}} \\ n=3: \frac{365}{365} \cdot \frac{364}{365} \cdot \frac{363}{365}=\frac{365 \cdot 364 \cdot 363}{365^{3}} \end{array}$$ (b) Use the pattern in part (a) to write an expression for the probability that four people $(n=4)$ have distinct birthdays. (c) Let $P_{n}$ be the probability that the $n$ people have distinct birthdays. Verify that this probability can be obtained recursively by $$P_{1}=1 \quad \text { and } \quad P_{n}=\frac{365-(n-1)}{365} P_{n-1}$$ (d) Explain why $Q_{n}=1-P_{n}$ gives the probability that at least two people in a group of $n$ people have the same birthday. (e) Use the results of parts (c) and (d) to complete the table. $$\begin{array}{|c|c|c|c|c|c|c|c|} \hline n & 10 & 15 & 20 & 23 & 30 & 40 & 50 \\ \hline P_{n} & & & & & & & \\ \hline Q_{n} & & & & & & & \\ \hline \end{array}$$ (f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than $\frac{1}{2} ?$ Explain.

Consider a group of $n$ people.
(a) Explain why the following pattern gives the probability that the $n$ people have distinct birthdays.
$$\begin{array}{l}
n=2: \frac{365}{365} \cdot \frac{364}{365}=\frac{365 \cdot 364}{365^{2}} \\
n=3: \frac{365}{365} \cdot \frac{364}{365} \cdot \frac{363}{365}=\frac{365 \cdot 364 \cdot 363}{365^{3}}
\end{array}$$
(b) Use the pattern in part (a) to write an expression for the probability that four people $(n=4)$ have distinct birthdays.
(c) Let $P_{n}$ be the probability that the $n$ people have distinct birthdays. Verify that this probability can be obtained recursively by
$$P_{1}=1 \quad \text { and } \quad P_{n}=\frac{365-(n-1)}{365} P_{n-1}$$
(d) Explain why $Q_{n}=1-P_{n}$ gives the probability that at least two people in a group of $n$ people have the same birthday.
(e) Use the results of parts (c) and (d) to complete the table.
$$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline n & 10 & 15 & 20 & 23 & 30 & 40 & 50 \\
\hline P_{n} & & & & & & & \\
\hline Q_{n} & & & & & & & \\
\hline
\end{array}$$
(f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than $\frac{1}{2} ?$ Explain.

Key Concepts

Recursive Formulation in Probability

The recursive formulation provides a systematic way to calculate the probability for n people based on the probability for n-1 people. By considering the effect of adding one more person, whose birthday must differ from all previous ones, the recurrence relation P? = [(365 - (n - 1)) / 365] · P??? is derived.

Complement Rule in Probability

The complement rule is used to find the probability of an event occurring by subtracting the probability of its complement from one. In this problem, to determine the probability that at least two people share the same birthday, one subtracts the probability that all birthdays are distinct (P?) from one, yielding Q? = 1 - P?.

Multiplication Principle in Probability

This principle explains that when making sequential independent choices, the overall probability of a sequence of outcomes is the product of the probabilities of the individual outcomes. In the context of the problem, when each person’s birthday is considered one after another, each person has fewer available days than the previous one, resulting in a decreasing product of probabilities.

Counting Method in Combinatorics

The method involves counting the number of favorable outcomes and dividing by the total number of possible outcomes. For distinct birthdays, the favorable outcomes are determined by the number of ways to assign different birthdays to each person, represented as a product of decreasing numbers, while the total number of outcomes is given by raising the total days (365) to the power of the number of people.

Transcript

00:01 We have to consider a group of n people.

00:03 Explain why the people pattern given the probability that the n people have distinct vertices.

00:09 So the pattern given the probability that n people have distinct vertex in distinct birth rates.

00:23 In the probability you can write this p of e is equal to n of e by n of s.

00:29 The sample space is the total number of possible combinations.

00:33 Total combinations.

00:39 Since person could be born on any of the 365 days, this number is 365 per n.

00:47 Enough is as 65 power n because there can be n number of possibilities.

00:54 In order for all these people to have different birthdays, the first person could be born on any of the 365 days of the year.

01:02 The second should be born on 364 year.

01:05 The third should be won 363 and so on.

01:07 So, n of e, the first person should be born in 365 days, second, 364, 3363, and so on.

01:18 So, if we can write p of e will be equal to n of e by n of s, which is equal to 365 into 364 into so on into 365 minus n minus 1 by 365 power x.

01:41 Second question we have to find the probability that amongst four that four people have just things but this so then n is equal to four we enright 365 by 365 then 364 by 365 363 by 365 this is 365 this is 365 into 365 4 into 363 into 362 by 365 whole power 4.

02:26 Next question they are asking us if p of n be the probability that n people have the strength birth with verify that p of 1 of 1 and p of n.

02:39 Here we can use the probability of 4 people having different birthdays as an example so p of 4 will be nothing but 364 minus 4 minus 1 by 365 into p of 4 minus 1 we're taking n is equal to 4 as an example here.

03:02 So this will be nothing but 362 by 365 into p of 3.

03:07 So that is nothing but 362 by 365 into 365 minus 3 minus 3 minus 3.

03:16 1 by 365 into p of 2 which is 362 into 362 into 363 by 365 into p of 2 p of 2 can be again expanded as 363 into 362 by 365 into 365 minus 2 minus 1 by 365 into p of 1...

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