Question

An integer is chosen between 70 and 100, Find the probability that it is
a. A prime number
b. Divisible by 7

Answer 1

Hint: In this type of question we have to use the concept of probability. Here, we have to find the probability of that chosen integer which is a prime number as well as for that chosen integer which is divisible by 7. We know that the probability for any event is given by, \[\text{Probability = }\dfrac{\text{Number of favourable outcomes}}{\text{Number of total outcomes}}\].

Complete step-by-step answer:
Now, we have to find the probability of an integer chosen between 70 to 100 which is a prime number as well as a chosen integer which is divisible by 7.
Here we have given two events so let us consider
A be the event of prime numbers between 70 to 100 and
B be the event of integers which are divisible by 7 between 70 to 100.
Now, we know that Number of integers between 70 to 100: \[\left\{ 71,72,73,74,..............,97,98,99 \right\}\]
So we get total number of outcomes = 29
Also we have the prime numbers between 70 to 100 which are: \[\left\{ 71,73,79,83,89,97 \right\}\]
So we get the number of favourable outcomes to an event A = 6
Also we can list the numbers divisible by 7 between 70 to 100 which are: \[\left\{ 77,84,91,98 \right\}\]
Hence, we get the number of favourable outcomes to an event B = 4
Also we know that, the probability for any event is given by,
\[\text{Probability = }\dfrac{\text{Number of favourable outcomes}}{\text{Number of total outcomes}}\]
Thus, we can find out the required probabilities as follows:

a. Probability that the chosen integer is a prime number i.e.
\[\Rightarrow \text{P}\left( \text{A} \right)\text{ = }\dfrac{\text{Number of favourable outcomes}}{\text{Number of total outcomes}}\]
\[\Rightarrow \text{P}\left( \text{A} \right)\text{ = }\dfrac{6}{\text{29}}\]

b. Probability that the chosen integer is divisible by 7 i.e.
\[\Rightarrow \text{P}\left( \text{B} \right)\text{ = }\dfrac{\text{Number of favourable outcomes}}{\text{Number of total outcomes}}\]
\[\Rightarrow \text{P}\left( \text{B} \right)\text{ = }\dfrac{4}{29}\]
Hence, the probability that the chosen integer between 70 to 100 is a prime number is \[\dfrac{6}{\text{29}}\] and the probability that the chosen integer between 70 to 100 is divisible by 7 is \[\dfrac{4}{29}\].

Note: In this type of question students have to know the formula of probability. Also they must identify the event from the question so it will be easy to find the corresponding probability.