Binomial distribution

I am tad stuck with part b of this question , I mainly don’t get what i need to for “ more than half of the rolls part”...

The random variable X represents the score after a single roll of a biased six-sided die for whichP (X=n)= kn for n =1,2,3,4,5,6 and constant k.(a) Find the value of the constant k. (b) The die is rolled 20 times. Calculate the probability that the score is greater than 4 on more than half of the rolls.

So I tried to find the probability of P(X>4) for a single roll using kn from part a , I worked out the k to be 1/21. But I am not sure what to do next ...
How do I take into account the phrase “more than half of the rolls “ ?

Will much appreciate the help !

Reply 1

vc94

Let Y=no. of times score is greater than 4 from 20 rolls.
This is a binomial distribution Y~B(20, p) where p=P(X>4).
“more than half of the rolls “ means Y>10.

Reply 2

starlit03

Original post by vc94

Let Y=no. of times score is greater than 4 from 20 rolls.
This is a binomial distribution Y~B(20, p) where p=P(X>4).
“more than half of the rolls “ means Y>10.

Ohhhh I see than u can do binomial cumulative distribution, wow that is really clever
I didn’t see that ! I didn’t even think to add make another variable Y aha
Thank you ! It makes sense .

Reply 3

starlit03

Original post by vc94

Let Y=no. of times score is greater than 4 from 20 rolls.
This is a binomial distribution Y~B(20, p) where p=P(X>4).
“more than half of the rolls “ means Y>10.

Can I ask , how did you know to make another variable, are there like any clues in the question that tell you ?
Again thank you

Reply 4

vc94

Mostly experience! Just try to identify a situation where you have repeated trials and a constant probability of success.
Sometimes you have a binomial situation where the p probability comes from a Normal Distribution probability.

Reply 5

swift1188

would anyone be able to help me workout part (a) of this question?

Reply 6

Erturgul_returns

The probability of the observed values of n must add to 1 In the question it's says n=knSo u jus substitute the values for n and get k , 2k and so on and make it equal to one
And find K

(edited 2 years ago)

Reply 7

panda34678

Original post by vc94

Let Y=no. of times score is greater than 4 from 20 rolls.
This is a binomial distribution Y~B(20, p) where p=P(X>4).
“more than half of the rolls “ means Y>10.

hi im also struggling on this question and dont understand how to work out adding a second variable? could you help?

Reply 8

mqb2766

Original post by panda34678

hi im also struggling on this question and dont understand how to work out adding a second variable? could you help?

What are you struggling with? The question gives the probabilities of each number on a die occurring and asks about the number of times a score >=4 occurs.

You use the given information to get the cumulative probablity P(X > 4) where X is the die score random variable, so the frequency or (cumulative) probabililty that the score is >4 for a single die. Then you want to calculate the probability that this occurs (or not, so a binomial distribution) more than 1/2 the times in 20 rolls so the probablity that it happens 11 or more times. This is a standard (cumulative) binomial where p=P(X>4) and n=20 and P(Y>11) where Y is the random variable representing the number of occurrences.

(edited 12 months ago)

Reply 9

panda34678

Original post by mqb2766

ahh tyyy!!
that makes so much more sense

Reply 10

mqb2766

Original post by panda34678

ahh tyyy!!
that makes so much more sense

NP, its a reasonably common question where they want you to calculate a cumulative probability value (normal, binomial, discrete), then ask something about the number of times it occurs so a binomial distribution using p from the first part.