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I think this is a variation on calculating at least one event, so may be related to the complement rule, or perhaps the poisson process.

Suppose I am observing my friend, waiting for him to blink. The probability that he will blink at least once in 1 second is constant, denoted by $p$.

Now I observe him for a variable interval, denoted by $t$. What is the probability that I will observe him blink at least once in $t$ second(s)?

sammosummo
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  • What's the chance he won't blink for one second? And for two seconds? And for $t$ seconds? And how does this lead to the required answer? – Alexander Geldhof Oct 23 '19 at 19:34

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$P($Friend doesn't blink in $t$ seconds$) = [P($Friend doesn't blink in $1$ second$)]^t = (1-p)^t$

So $P($Friend blinks at least once in $t$ seconds$) = 1-P($Friend doesn't blink in $t$ seconds$) = 1-(1-p)^t$

With these types of questions, you usually have to look at the complement. "At least once" should make bells go off in your brain. It means the exact same thing as "not $0$ times" (assuming we are talking about something countable, like blinks, which can only be integers $\ge 0$). To calculate this, you first have to figure out what the probability is of this happening exactly $0$ times and then take the complement.

yagod
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    If $p=0.1$ and $t=1$, then $P(\ge 1\textrm{ blink in }t)=p=0.1$. But according to your solution, $P(\ge 1\textrm{ blink in }t)=1-p^t=1-0.1^1=0.9$ – sammosummo Oct 23 '19 at 19:40
  • whoops, sorry, I mistakenly thought $p$ was the probability that he doesn't blink in one second. I corrected it now. – yagod Oct 23 '19 at 19:47