Question

Given the probability $p = \dfrac{{87}}{{100}}$ that an event will not happen, how do you find the probability that the event will happen ?

Answer 1

Hint:We need to find the probability that the event E will happen using some basic probability formula. The given events in this problem are mutually exclusive and they cannot happen simultaneously. Use the formula $P(E) + P(\bar E) = 1$
Where, $P(E)$ is the probability that event E will occur and $P(\bar E)$ is the probability that the event E will not occur.
Then make rearrangement in the formula to get $P(E)$ and substitute the given values and solve the problem to get the required probability.

Complete step by step solution:
Let us first denote the given terms.
Let $P(E)$ denotes the probability that an event E will happen.
Let $P(\bar E)$ denote the probability that an event E will not happen.
Given $P(\bar E)= \dfrac{{87}}{{100}}$.
Now we know that the sum of all the probabilities of all possible mutually exclusive events of an experiment is 1.
Here we note that the event E either happens or not happens. Also these are two mutually exclusive events.
Since it cannot be that the event E happens and not happens at the same time.
Hence we can say that the probability of E + the probability of not E is equal to 1.
.i.e. $P(E) + P(\bar E) = 1$ ……(1)
In the given problem we need probability if an event E will happen.
So we make rearrangement in the equation (1) to get $P(E)$.
We take $P(\bar E)$ to R.H.S. in the equation (1), we get,
$P(E) = 1 - P(\bar E)$
$ \Rightarrow P(E) = 1 - \dfrac{{87}}{{100}}$
Now taking LCM in the R.H.S. we get,
$ \Rightarrow P(E) = \dfrac{{100 - 87}}{{100}}$
$ \Rightarrow P(E) = \dfrac{{13}}{{100}}$
Hence the probability that an event E will happen is $\dfrac{{13}}{{100}}$.

Note:
Remember that the result of the sum of mutually exclusive probabilities is one if all the cases are considered, that is all mutually exclusive possible events.
i.e. $P(E) + P(\bar E) = 1$.
For example, if a coin is tossed then there are two mutually exclusive events that are getting a head and tail. Hence the probability of getting head + probability of getting tail is 1.
Also note that the probability of an event E will always be in between 0 and 1.