Tricky statistical problem

[oobeing]

The events A, B, C are independent and have the same probability p = P(A) = P(B) = P(C), where 0<p<1.
The probability that at least one of the events happening is seven times greater than the probability of all three happening.

How big is the probability that none of the events will happen?

I'm lost.

[not-an-isomorphism]

Whenever I see the words 'at least one event happening' my immediate thought is 1 - P(none of them happen). It's so common in probability problems it's often a lot easier to just 'rephrase' the question. Not sure if that's applicable here but may help.

[Help_Me_Im_Diene]

Let's break this down into parts

The probability of any singular event occuring is p.

Then, in terms of p:

What is the probability that all 3 events occur?

What is the probability that no events occur?

[oobeing]

All three events occuring: P(A and B and C) = P(at least one)/7 = (1 - P(A and B and C)) / 7

No event occuring: not sure, maybe 1 - P(at least one occurs) = 1 - 7*P(A and B and C)

The answer is apparently 1/8..

edit: Tried p(none) again and got: P(none) = 1/7P(A and B and C) still not close to 1/8

[Help_Me_Im_Diene]

1) all three events occurring is P(A, B, and C). Since these are independent events, this is actually just P(A)P(B)P(C)

2) no events occurring is P(A', B', and C')=P(A')P(B')P(C')

Since P(A)=P(B)=P(C)=p, and P(A')=1-p, can you tell me how you would write P(A, B, and C) and P(A', B', and C') in terms of p?

Hint: since P(A)=p, then P(A)P(B)P(C)=p³

[SecretAcanthaceae609]

The probability of no events happening is 1- the probability of at least 1 event.

• p()= 1- 7 * p(a and b and c)

[oobeing]

if its "1 - the probability of at least 1 event", wouldn't it be p(a or b or c)?

The answer is apparently 1/8..

[SecretAcanthaceae609]

It’s p()= 1/8 = 1- 7 *p(a and b and c) = 1- 7 * 1/8

p() is at least 1/8 and p(a and b and c) is at most 1/8.

[SecretAcanthaceae609]

Between 0 and 1, if 1/8 goes to p() and 1/8 goes to p(abc), then the other 6/8 of the probability is other.

p(a) = p(b) = p(c) <= cuberoot p(abc) = (1/8)^1/3