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

12

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

OP

14

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

OP

14

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

12

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

3

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

Reply 6

Erturgul_returns

1

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 1 year ago)

Reply 7

panda34678

6

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

21

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 3 months ago)

Reply 9

panda34678

6

Original post by mqb2766

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.

ahh tyyy!!
that makes so much more sense

Reply 10

mqb2766

21

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.

Binomial distribution

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