S2 Conditions for Distribution

What is the difference between saying probability of success in each trial is same and each probability of success is independent of the others? Surely the second is covered by the first? (Binomial Distribution)

(edited 7 years ago)

Reply 1

Zacken

Original post by Oblogog

No, the probability of the first trial could be 1/2 and the probability of the second trial could be 1/3 and they could be independent, but have different probabilities.

Reply 2

Gregorius

Original post by Oblogog

This is a linguistic thing. Strictly speaking one should say that two random variables are dependent/independent of one another rather than saying that two probabilities of success are dependent/independent of one another. As @Zacken has pointed out, it is perfectly possible for two binomial random variables to be independent of one another, but their probabilities of success to be different.

Just to complicate matters, it is perfectly possible to have two binomial random variables where the probability of success for each of them is itself a random variable. Then one could talk about the probabilities of success being dependent or independent.

Reply 3

Oblogog

Original post by Zacken

No, the probability of the first trial could be 1/2 and the probability of the second trial could be 1/3 and they could be independent, but have different probabilities.

What I'm saying is if probability is constant then it implies independent not the other way around. Yet the mark scheme (from CGP book I'm using) likes you to say both instead of just the first.

(edited 7 years ago)

Reply 4

Zacken

Original post by Oblogog

What I'm saying is if probability is constant then it implies independent not the other way around

Not so, a trivial counterexample is that if an event A occurs with probability p, then it is not independent to itself, obviously... yet it has the same probability of occurs as itself of occurring.

Reply 5

Oblogog

Original post by Zacken

Not so, a trivial counterexample is that if an event A occurs with probability p, then it is not independent to itself, obviously... yet it has the same probability of occurs as itself of occurring.

I don't understand that. Why can't I say "no matter what the probability of x happening is p' and hit 2 birds with one stone?

Reply 6

Zacken

Original post by Oblogog

I don't understand that. Why can't I say "no matter what the probability of x happening is p' and hit 2 birds with one stone?

That's because having the same probability has nothing to do with being independent.

I could pass my exam with probability 1/2 and feel happy with probability 1/2, but passing my exams and feeling happy don't need to be independent events even if they have the same probability.

So a binomial distribution says that each trial is independent of the other and the probability of each trial is the same, those two conditions are different.

Reply 7

Oblogog

Original post by Zacken

That makes what I said about probability being constant implying independent wrong (bad use of words) but I don't think it makes what I said about 'same probability no matter what' wrong? By which I mean if you didn't pass exam the happiness probability might change, meaning it doesn't fit the condition same probability no matter what.