Probability Question

Russell is an amateur snooker player. He practises a particular pot several times and the probability of
making the pot each time is 0.3. In any one practice session, his goal is to make the pot 8 times. You may
assume that his probability of making the pot remains constant and all attempts at the pot are
independent.
In one week, he has 25 practice sessions and achieves his goal each time.
Estimate the probability that his average number of shots in each session was less than 25.

I assumed this was a negative binomial with X~NB(8,0.3) as probability of success is 0.3 and he needs to pot it 8 times. I assume that I’m meant to find P(X is less the or equal to 24) but I don’t have a formula for cumulative negative binomial and when I tried adding up the probabilities it was quite far off. The answer is 0.145. I’m not sure where I have gone wrong with this. Any help would be appreciated. Thanks
Haven't really pencil down anything, but what you're supposed to find is this:

$X_{n}$: RV of the number of shots in the nth session, so $X_{n} \sim NB(8, 0.3)$, and
$T_{25} = \sum^{25}_{i=1}X_{i}/25$, with each $X_{i}$ being iid (independent and identically distributed).
Our goal is to find $P(T_{25}<25)$.
(P.S. I put subscripts here, because you can generalize this)

So some sessions could take more than 25 shots, but on the whole you need less than 25 on average.
This is hard to find (maybe it's easy, but I'm too sleepy to figure it out...).

But rephrase this question a bit, this is essentially asking for "less than 25 attempts on average in 25 sessions with 8 successes each session", which is the same as "less than 25*25 attempts with 8*25 successes in total" (why?). The latter is easier to manage. Granted I think you still need the CMF of negative binomial...
(edited 2 weeks ago)
Original post by tonyiptony
Haven't really pencil down anything, but what you're supposed to find is this:

$X_{n}$: RV of the number of shots in the nth session, so $X_{n} \sim NB(8, 0.3)$, and
$T_{25} = \sum^{25}_{i=1}X_{i}/25$, with each $X_{i}$ being iid (independent and identically distributed).
Our goal is to find $P(T_{25}<25)$.
(P.S. I put subscripts here, because you can generalize this)

So some sessions could take more than 25 shots, but on the whole you need less than 25 on average.
This is hard to find (maybe it's easy, but I'm too sleepy to figure it out...).

But rephrase this question a bit, this is essentially asking for "less than 25 attempts on average in 25 sessions with 8 successes each session", which is the same as "less than 25*25 attempts with 8*25 successes in total" (why?). The latter is easier to manage. Granted I think you still need the CMF of negative binomial...

Note also that "X successes in less than N attempts" has the same probability (is the same event, basically) as "at least X successes in N-1 attempts", at which point you should be in familiar territory.

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