# Comment

### Comments

Comments in green were written by me. Comments in blue were not written by me.

@Matthew: Here is how I calculated it:

You want a specific set of 20 stickers. Imagine you have already \(n\) of these. The probability that the next sticker you buy is one that you want is

$$\frac{20-n}{682}.$$

The probability that the second sticker you buy is the next new sticker is

$$\mathbb{P}(\text{next sticker is not wanted})\times\mathbb{P}(\text{sticker after next is wanted})$$

$$=\frac{662+n}{682}\times\frac{20-n}{682}.$$

Following the same method, we can see that the probability that the \(i\)th sticker you buy is the next wanted sticker is

$$\left(\frac{662+n}{682}\right)^{i-1}\times\frac{20-n}{682}.$$

Using this, we can calculate the expected number of stickers you will need to buy until you find the next wanted one:

$$\sum_{i=1}^{\infty}i \left(\frac{20-n}{682}\right) \left(\frac{662+n}{682}\right)^{i-1} = \frac{682}{20-n}$$

Therefore, to get all 682 stickers, you should expect to buy

$$\sum_{n=0}^{19}\frac{682}{20-n} = 2453 \text{ stickers}.$$

You want a specific set of 20 stickers. Imagine you have already \(n\) of these. The probability that the next sticker you buy is one that you want is

$$\frac{20-n}{682}.$$

The probability that the second sticker you buy is the next new sticker is

$$\mathbb{P}(\text{next sticker is not wanted})\times\mathbb{P}(\text{sticker after next is wanted})$$

$$=\frac{662+n}{682}\times\frac{20-n}{682}.$$

Following the same method, we can see that the probability that the \(i\)th sticker you buy is the next wanted sticker is

$$\left(\frac{662+n}{682}\right)^{i-1}\times\frac{20-n}{682}.$$

Using this, we can calculate the expected number of stickers you will need to buy until you find the next wanted one:

$$\sum_{i=1}^{\infty}i \left(\frac{20-n}{682}\right) \left(\frac{662+n}{682}\right)^{i-1} = \frac{682}{20-n}$$

Therefore, to get all 682 stickers, you should expect to buy

$$\sum_{n=0}^{19}\frac{682}{20-n} = 2453 \text{ stickers}.$$

Matthew

on /blog/56

on /blog/56

@Matthew: Here is how I calculated it:

You want a specific set of 20 stickers. Imagine you have already \(n\) of these. The probability that the next sticker you buy is one that you want is

$$\frac{20-n}{682}.$$

The probability that the second sticker you buy is the next new sticker is

$$\mathbb{P}(\text{next sticker is not wanted})\times\mathbb{P}(\text{sticker after next is wanted})$$

$$=\frac{662+n}{682}\times\frac{20-n}{682}.$$

Following the same method, we can see that the probability that the \(i\)th sticker you buy is the next wanted sticker is

$$\left(\frac{662+n}{682}\right)^{i-1}\times\frac{20-n}{682}.$$

Using this, we can calculate the expected number of stickers you will need to buy until you find the next wanted one:

$$\sum_{i=1}^{\infty}i \left(\frac{20-n}{682}\right) \left(\frac{662+n}{682}\right)^{i-1} = \frac{682}{20-n}$$

Therefore, to get all 682 stickers, you should expect to buy

$$\sum_{n=0}^{19}\frac{682}{20-n} = 2453 \text{ stickers}.$$

You want a specific set of 20 stickers. Imagine you have already \(n\) of these. The probability that the next sticker you buy is one that you want is

$$\frac{20-n}{682}.$$

The probability that the second sticker you buy is the next new sticker is

$$\mathbb{P}(\text{next sticker is not wanted})\times\mathbb{P}(\text{sticker after next is wanted})$$

$$=\frac{662+n}{682}\times\frac{20-n}{682}.$$

Following the same method, we can see that the probability that the \(i\)th sticker you buy is the next wanted sticker is

$$\left(\frac{662+n}{682}\right)^{i-1}\times\frac{20-n}{682}.$$

Using this, we can calculate the expected number of stickers you will need to buy until you find the next wanted one:

$$\sum_{i=1}^{\infty}i \left(\frac{20-n}{682}\right) \left(\frac{662+n}{682}\right)^{i-1} = \frac{682}{20-n}$$

Therefore, to get all 682 stickers, you should expect to buy

$$\sum_{n=0}^{19}\frac{682}{20-n} = 2453 \text{ stickers}.$$

Matthew

on /blog/56

on /blog/56

2019-05-29You want a specific set of 20 stickers. Imagine you have already \(n\) of these. The probability that the next sticker you buy is one that you want is

$$\frac{20-n}{682}.$$

The probability that the second sticker you buy is the next new sticker is

$$\mathbb{P}(\text{next sticker is not wanted})\times\mathbb{P}(\text{sticker after next is wanted})$$

$$=\frac{662+n}{682}\times\frac{20-n}{682}.$$

Following the same method, we can see that the probability that the \(i\)th sticker you buy is the next wanted sticker is

$$\left(\frac{662+n}{682}\right)^{i-1}\times\frac{20-n}{682}.$$

Using this, we can calculate the expected number of stickers you will need to buy until you find the next wanted one:

$$\sum_{i=1}^{\infty}i \left(\frac{20-n}{682}\right) \left(\frac{662+n}{682}\right)^{i-1} = \frac{682}{20-n}$$

Therefore, to get all 682 stickers, you should expect to buy

$$\sum_{n=0}^{19}\frac{682}{20-n} = 2453 \text{ stickers}.$$

on /blog/56