Question

Cards marked with the number 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from the box. Find the probability that the number on the card is:
1. An even number
2. A number less than 14
3. A number is perfect square
4. A prime number less than 20

Answer 1

Hint: Find the total number of favorable outcomes in each case and then use the basic concept of probability i.e. \[{\text{Probability}} = \dfrac{{No.{\text{ }}of{\text{ }}Favourable{\text{ }}Outcomes}}{{Total{\text{ }}No.{\text{ }}of{\text{ }}Outcomes}}\] here to find the probability of each case.

Complete step-by-step answer:
 We know that the formula of probability is given as; \[{\text{Probability}} = \dfrac{{No.{\text{ }}of{\text{ }}Favourable{\text{ }}Outcomes}}{{Total{\text{ }}No.{\text{ }}of{\text{ }}Outcomes}}\]
where favorable outcomes define the outcomes of interest

According to the given information in a box cards marked with numbers 2 to 101 are mixed thoroughly
Since, it is given that in the cards marked with numbers 2 to 101 which means first card has number 2 and last has number 100
So, the total numbers of outcomes are 100
Case 1: The probability that the number of cards is an even number.
 According to the question, the number on the cards are 2 to 101
We know that even numbers given as; 2n where n is integer
Thus the even numbers between the cards from 2 to 101 are 2,4,6,8, 10, ….,100 = 50
So here no. of favorable outcomes is equal to 50 and total number of outcomes is 100
Probability that the card drawn is an even number = $\dfrac{{No.{\text{ }}of{\text{ }}Favourable{\text{ }}Outcomes}}{{Total{\text{ }}No.{\text{ }}of{\text{ }}Outcomes}}$ = $\dfrac{{50}}{{100}} = \dfrac{1}{2}$
Case 2: The probability that the number on the card is less than 14.
The numbers on the card less than 14 are 2, 3, 4, 5, 6, …., 13 = 12
Therefore, Number of favorable outcomes = 12
So Probability = $\dfrac{{No.{\text{ }}of{\text{ }}Favourable{\text{ }}Outcomes}}{{Total{\text{ }}No.{\text{ }}of{\text{ }}Outcomes}} = \dfrac{{12}}{{100}} = \dfrac{3}{{25}}$
Case 3: The probability that the number on the card is a perfect square.
We know that perfect square are integers which are square of some integer
The perfect square numbers on the cards are perfect square are 4, 9, 25, 36, 49, 64, 81, 100
therefore, Number of favorable outcomes = 9
So Probability = $\dfrac{{No.{\text{ }}of{\text{ }}Favourable{\text{ }}Outcomes}}{{Total{\text{ }}No.{\text{ }}of{\text{ }}Outcomes}}$ = $\dfrac{9}{{100}}$
Case 4: The probability that the number on the card is less than 20.
The numbers less than 20 on the cards are 2,3,5,7,11,13,17,19
therefore, Number of favorable outcomes = 8
So Probability = $\dfrac{{No.{\text{ }}of{\text{ }}Favourable{\text{ }}Outcomes}}{{Total{\text{ }}No.{\text{ }}of{\text{ }}Outcomes}}$ = $\dfrac{8}{{100}} = \dfrac{2}{{25}}$

Note: The probability theory provides a means to get an idea of the likelihood that different events will occur as a result of a random experiment in terms of quantitative measures ranging from zero to one. The chance for an unlikely or impossible occurrence is zero, and one for an occurrence that is likely or certain to happen.