probability
During a game of musical chairs, 10 people walk around eight chairs waiting for the music to stop. Find the probability of a person not getting a chair.
I have a hard time understanding probability to be honest. I thought the answer to this would simply by $\frac{2}{10}$ but I feel like I might need to use combinations for this, however I can't figure out how I will do that.
Although there is surely a closed-form solution as @BGM alluded to, you can also simulate the situation. Following is code to simulate a sampling distribution.
set.seed=126790
sims=100
probs=numeric(sims)
for(k in 1:sims)
{
reps=1000
rolls=matrix(NA,nrow=reps,ncol=8)
for(i in 1:reps)
{
dice=c(NULL)
for(j in 1:8)
{
dice=c(dice,sample(c('a','a','b','c','d','e'),1))
}
dice[sample(1:8,1)]=sample(c('a','a','b','c','d','e'),1)
rolls[i,]=dice
}
rolls
counter=0
for(i in 1:reps)
{
if(length(grep('a',rolls[i,]))==6||
length(grep('b',rolls[i,]))==6||
length(grep('c',rolls[i,]))==6||
length(grep('d',rolls[i,]))==6||
length(grep('e',rolls[i,]))==6){
counter=counter+1
}
}
probs[k]=counter/reps
cat(k,'\n')
}
probs
mean(probs)
sd(probs)
hist(probs)
In my run of the simulation, the mean probability of getting a 6 character match was $1.959$% with standard deviation $0.397$%. Because the standard error is the standard deviation of the sampling distribution, in 95% of repetitions of 100 sets of 8 rolls with one being replaced, between 1.18% and 2.74% will have a 6-character match. Of course this assumes a random replacement roll, but most people would try to behave strategically. However, I doubt it would change the results by very much. Additionally, this assumes the rolls are normally distributed. Visual analysis says the rolls are normally distributed.
The first part is right.
For the simulation, are you also counting the case where the chosen numbers are $1$ and $N$?
Best Answer
The unstated assumption is that all the people are equally likely to get a chair. Then you are correct that the chance a particular person does not get a chair is $\frac 2{10}$. The chance that some person does not get a chair (which is another way of reading what you wrote) is $1$.