Anyone good at maths? Calculating probability

[Monoceros]

Can anyone please help with the following calculation:
143 surveys per year
307 people

I know that the probability of each person being surveyed is 47% (143/ 307)

But what's the probability of each person being surveyed:
Twice
Three times
Four times
Five times per year?

Does the probability just get divided by 2, i.e.
Probability of one person being surveyed twice in a year is 47%/ 2 = 23.5%???

I would be grateful for your help with this calculation.

It's B. She wants to know roughly how many people would have completed 2 (or more) surveys by the end of the year.

I think if I am reading correctly that the surveys also have various possible outcomes too, so they are more ratings than surveys? Completed by a third party?

Because if you're looking to reward people who have had (x) positive surveys then you also need to look at the threshold for it to be considered positive and also how people are typically allocating scores. For example the thread a few months ago about AirBnB ratings. People completing the rating process will have different ideas about what 5/5 means. And the person deciding the reward for positive surveys or penalty for negative ones needs to take this into account otherwise you get this issue like in AirBnB where the guest thinks that 4 is a good rating, but actually hosts are being penalised for not showing a 5.

Yes, @BertieBotts, correct, option 2.
I'm not focusing on different outcomes of the surveys, as the outcomes don't really matter. There are only 2 outcomes:

1. Positive (met standard)
2. Negative (didn't meet standard)

I'm just purely interested in a probability of an individual being surveyed 1, 2, 3, 4 and 5 times in a year if, on average, 143 surveys are conducted per year (11 per period of 4 weeks).

One cause of the confusion here is the issue of how the surveys are sent out. I think your OP implies something like

Survey system A: on Monday, send out 143 surveys to a random selection of people from the 307 eligible people. Each person gets 0 or 1 surveys. On Tuesday, do the same again and keep going.

That gives each person a decent chance of getting a survey.

But my maths is for this situation

Survey system B: on Monday, send 1 survey to 1 of 307 people and that's it. On Tuesday, send 1 survey again to 1 lucky person (could be the same person again). Repeat on Wednesday and keep going til you've sent 143 surveys.

Then asking if someone gets 2 surveys is like asking if they have won the lottery twice. And is pretty unlikely to happen.

If you want people to do multiple surveys, you've got to get many more sent out there. More chances to win the lottery.

@parietal it's option B
The odds indeed seem like winning the lottery and, now in possession of all the info and mathematical calculations - thanks to you all, I'll be contacting the survey providers to request a higher number of surveys in order to improve these odds.
Thank you all for your help.

If it's 11 every 4 weeks, are these in a batch, so each batch of 11 will all be assigned to different staff members, yes? No duplications in each batch.

So in theory it's possible (even though unlikely) for one person to receive two surveys four weeks apart? They have 13 opportunities to receive a survey every year, but only a 11/307 (about 3.6%) chance to receive one each time?

I think you've confused things and caused rounding errors with your 143 adding up

So actually the numbers in your first post aren't right at all.

The actual possibility of receiving at least 1 survey every year is about 48% - done by first calculating the chance of receiving no survey in any 4-week batch of 11 (96.4%) and then multiplying the 96.4% by 96.4% 13 times - this makes your initial calculation right.

Receiving at least 2 surveys is much lower. You do this by thinking about the chance of receiving one survey over 12 times - which is 35.6% (inverse of 64.4% which is 0.964 x 0.964, 12 times) multiplied by 48%. So 17%.

Receiving at least 3 surveys = 6%

Receiving at least 4 surveys = 2%

I've done rounding errors here too but you get the idea. There are online calculators which will do it for you properly.

I wouldn't say it's quite winning the lottery, but if it's supposed to be achievable by a majority, then you're very far off that.

That's not quite right @BertieBotts
The surveys are not done in batches of 11. The surveys are completely random (there could be 1 survey in Mon, 1 on Tue, 2 the following week Thursday etc), but we receive approximately 11 surveys every 4 weeks from mystery shoppers. So, in theory, it's possible, but very unlikely, for the same person to be the recipient of every single survey conducted.
I hope that makes sense!

Ah I see, sorry. Then I think Paretial has it right :)

I'd bet you that they're not completely random, unless the survey programme is driven by a random sampling algorithm.

How many surveys have been done this year? - and you've not had one duplicate.
How are the Mystery Shoppers assigned to the surveys?

Your automatic response has been to suggest more surveys to enhance the Gold Star reward scheme. This may not be the right response. The entire reward scheme appears very odd, and is almost certainly regarded as an unfair joke by anyone that's aware of it and has understanding of the parameters.

Yes, @DogInATent the whole system seems very unfair and unachievable.
I agree that the surveys might not be completely random but I have no insight into the allocation of mystery shoppers and their working practices. The only thing I can do is highlight the unfairness of this system and push for a change.
So far this year we've had about 85 surveys (our year runs from April) and no duplications at all.

The straight numbers game suggests it's very unfair.

But you need to know how the system works in detail to improve it, because the pure numbers game only works if it's genuinely random. With the data you have you can do two things:

• Demonstrate the system is fair/unfair based on the assumption of pure randomness.
• Prove or disprove with a confidence limit the hypothesis that the system is random or not.

By the way, this is an excellent example to disprove the idea of, "Why are we learning probability in maths, I'll never use it in real life"!

Hey OP, probably best to ask ChatGBT?