office dispute regarding probability

Question

Good afternoon,

My office is having a dispute involving statistics and I’m seeking your help. It’s an easy problem but wanted one of you to back up what I’m saying. So it involves a coffee game. Our boss writes a number 1-10 on the board and covers it. Whoever picks the number on the board buys coffee. There are 5 people (including myself) under my boss and he tells each of us to pick a number without revealing if anyone picked the number. So, as a group we pick 5 numbers (one number each) and if one of us picks it that person buys, if the number is one of his he buys (again not revealing until all 5 have picked). Who has the advantage (boss or group)? What’s the probability of each person in the group picking the wrong number?

Example from today… Boss has number covered, coworkers pick 7,3,6,4, and I pick 5. Boss says Matt you picked 3 that’s the number so you’re buying coffee. Can you give me some stats for that and save the day.

Answer 1

I will make the assumption that the Boss buys if no one picks the number that he wrote on the board.

 

In that case, he effectively has five chances to pay for coffee and each person has 1 chance out of 10.

 

So the probablity is that your Boss will pay 50% of the time and that each of the five of you will pay 10% of the time.

 

Another way to look at this is to say what is the probabilty that your numer will not match the hidden number

 

for the first person it you be 9/10 or 90%

for the 2nd person it you be 8/9 or  88.8%

for the 3rd person it would be 7/8 or 87.5%

for the 4th person it would be 6/7 or 85.7%

for the 5th person it would be 5/6 or 83.3%

 

now if you multiply these probabilties together you will get the probability that none of the five people match the number

 

when you multiply these percentages the result is 50%

 

so chances are that 50% of the time one of the workers will pay and 50% of the time the boss pays

 

so there is no advantage

Answer 2

Hi Steve--

 

seems to me that each of you 5 guys has a 1 in 10 shot of picking the wrong number, while boss man has a 5 in 10 of getting stuck with it.

 

But this is only at the start of the game.

 

Once it starts, say you pick first, and you pick a "safe" number, everyone else just got a little closer to picking the unlucky one. (Boss will now be 5 out of 9, lowly servants will be 1 out of 9.) If the next guy picks a safe number, then everyone else's odds then become out of 8, etc. If the first 4 of you pick safe, the 5th of you is going to have 1 out of 6 odds, and boss man will have 5 out of 6 odds.

 

Now if your boss man were truly AMAZING, he'd buy the coffee every friday, and a box o' donuts, too. C'MON boss, the boys have been working hard all week!

 

Have fun, fellas!

 

Matt in NY (ps hope you don't get in trouble for any of this!)