Question

A machine generates a two-digit number randomly. Find the probability that the number generated is either less than 25 or greater than 85.
A) $\dfrac{{27}}{{89}}$​
B) $\dfrac{{28}}{{89}}$
C) $\dfrac{{28}}{{90}}$
D) $\dfrac{{29}}{{90}}$

Answer 1

For solving this question, we should know about the basic concepts of probability.
Here the first step will be calculating the total number of outcomes possible. Then we find the desired number of outcomes in both the scenarios mentioned in the question and then use the definition of probability of the number generated with respect to the desired outcomes.

Complete step by step solution:
Here we are going to find out the total number of outcomes possible.
The number of outcomes would be 90 (that is, counting from 10 till 99).
Now, we count the desired number of outcomes. Since, the number generated should be either less than 25 or greater than 85.
Thus, the desired outcomes are – 15 (When the number generated is less than 25).
And, the desired outcomes are – 14 (When the number generated is greater than 85).
Total desired outcomes when the number generated should be either less than 25 or greater than 85 = 15 + 14 = 29
Probability = $\dfrac{{{\text{Desired number of outcomes}}}}{{{\text{Total number of outcomes}}}}$
Probability = $\dfrac{{29}}{{90}}$
Hence, the probability that the number generated is either less than 25 or greater than 85 is $\dfrac{{29}}{{90}}$

So, option (D) is the correct answer.

Note:
We can solve this problem by another approach i.e. -
Probability (number generated should be either less than 25 or greater than 85) + Probability (number is not less than 25 or greater than 85) = 1 - (1)
Now, we need to find the probability of the number generated should be either less than 25 or greater than 85. Thus, we can calculate the probability of the number is not less than 25 or greater than 85 and then subtract that probability from 1. In this case, also the number of outcomes is 90. The desired number of outcomes is 61 (that is 25-85).
Thus, calculating the probability for number is not less than 25 or greater than 85 is-
Probability = $\dfrac{{{\text{Desired number of outcomes}}}}{{{\text{Total number of outcomes}}}}$
Probability = $\dfrac{{61}}{{90}}$
Now, putting in expression (1), we get,
$\Rightarrow \dfrac{{61}}{{90}}$+ Probability (number generated should be either less than 25 or greater than 85) = 1
Probability (number generated should be either less than 25 or greater than 85) = $\dfrac{{29}}{{90}}$
Hence, we derived the same result by this method also.