# Probability of an event occuring at least once in 50 tries

Probability of an event is $.116$. In $50$ tries, what are the chances at least one event occurs? I see that the probability that it wouldn't happen in one try is $.884$ and the probability that it wouldn't happen in two tries is $(.884)^2$.

• In questions like this, ask yourself: What are the chances that it does not happen? Commented Jul 12, 2015 at 18:53
• .884 for one try, (.884)E2 for 2, etc?
– Jim
Commented Jul 12, 2015 at 18:57
• Not quite. $1-0.884^{2}$, etc... Commented Jul 12, 2015 at 18:58
• @Jim, you answered my question correctly (assuming that your E2 notation means $.884^2$). So the answer to your original question will be ... Commented Jul 12, 2015 at 18:59

If an event occurs $0$ times (out of $50$, in this case) then it does not occur at least once. So we can find the probability of it not occurring and then subtract that value from $1$.

So, what are the chances of it not occurring on $1$ trial? $1-.116 = .884$

What about not occurring on $2$ trials? $(1-.116)\cdot (1-.116)=.884 \cdot .884 = .781456$

Now what about not occurring on $50$ trials? $.884^{50} \approx .002102$

We must subtract this value from $1$ (recall that what we just calculated is the probability of it not occurring, so the probability of it occuring at least once is: $$1-.002102 \approx .9979$$

A more general approach to this problem is through the binomial distribution.

If $$p$$ is the probability of success of an event, then the probability that we have exactly $$k$$ successes over $$n$$ independent trials is: $$P(X = k) = \dbinom{n}{k} \cdot p^k \cdot (1-p)^{n-k},$$ where $$X$$ is the random variable that expresses the number of successes of the event.

In our specific problem, we have that $$p= 0.116, n = 50$$ and we want $$P(X\ge 1) = P(X = 1) + P(X = 2)+ \cdots + P(X = 50) = 1-P(X = 0)$$

The chances of the event not happening after 1 trial is (1-0.116) = 0.884

The chances of the event not happening after 2 trials is $0.884^2$

The chances of the event not happening after 50 trials is

0.002101905........... ($0.884^{50}$)

Therefore, the chances of the event happening at least once is 1 minus the above number which is

0.99789880941.........

There is a 99.79% (to 4sf) chance the event will happen at least once in 50 trials.