Question

Cards numbered 1 to 30 are put in a bag. A card is drawn at random. Find the probability that the card drawn is:
(a) Not divisible by 3
(b) A prime number greater than 7
(c) Not a perfect square number

Answer 1

Here we use the method of probability to find the probabilities in each situation.
(a) Here we find the numbers that are divisible by three by dividing the largest number on the card by 3. Then subtract the number of cards from the total number of cards. Use a formula for probability to find the probability of a card drawn.
(b) Here we write the prime numbers after 7 and count the number of prime numbers. Use the formula for probability to find the probability of card drawn.
(c) Here we write the perfect squares lying between the numbers 1 to 30. Count the number of perfect squares and deduct that number from the total number of cards. Use a formula for probability to find the probability of a card drawn.

Complete step-by-step answer:
We are given cards numbered from 1 to 30 and put in a bag.
So, total number of cards in the bag $= 30$
Now one card is drawn at random. We solve for each case separately.
(a) Not divisible by 3
We find the number of cards that have a number that is not divisible by 3
Since, we know there are 30 numbers from 1 to 30.
We count the numbers from 1 to 30 that are divisible by 3
Numbers divisible by 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27 and 30.
So, number of cards divisible by 3 $= 10$
So, number of cards not divisible by 3 $=$Total number of cards $-$number of cards divisible by 3
$\therefore$Number of cards not divisible by 3 $= 30 - 10$
$\therefore$Number of cards not divisible by 3 $= 20$
Probability is given by the number of favorable outcomes divided by total number of observations.
Here the total number of observations is 30 and number of favorable outcomes is 20.
$\Rightarrow$ Probability $= \dfrac{{20}}{{30}}$
Cancel the same factor from numerator and denominator.
$\Rightarrow$Probability $= \dfrac{2}{3}$
(b) A prime number greater than 7
We find the number of cards that have a number that is a prime number but greater than 7.
We know from the definition of prime numbers that they are the numbers that have only 1 and the number itself as their factor.
Prime numbers from numbers after 7 till 30 are:
11, 13, 17, 19, 23 and 29
So, number of cards having a prime number greater than 7 $= 6$
Probability is given by the number of favorable outcomes divided by total number of observations.
Here the total number of observations is 30 and number of favorable outcomes is 6.
$\Rightarrow$ Probability $= \dfrac{6}{{30}}$
Cancel the same factor from numerator and denominator.
$\Rightarrow$Probability $= \dfrac{1}{5}$
(c) Not a perfect square number
We find the number of cards that have a number that is not a perfect square.
We know from the definition of perfect square that a number is said to be a perfect square if it is a square of a whole number. We write squares of whole numbers starting from 1 and continue till we get square of number greater than 30
We have${1^2} = 1,{2^2} = 4,{3^2} = 9,{4^2} = 16,{5^2} = 25,{6^2} = 36$
So, we neglect the value 6 as its square is greater than 30.
Numbers that are a perfect square from 1 to 30 are:
1, 4, 9, 16 and 25
So, number of cards that are perfect squares $= 5$
So, number of cards that are not a perfect square $=$Total number of cards $-$number of cards that are a perfect square
$\therefore$Number of cards that are not a perfect square $= 30 - 5$
$\therefore$Number of cards that are not a perfect square $= 25$
Probability is given by the number of favorable outcomes divided by the total number of observations.
Here the total number of observations is 30 and number of favorable outcomes is 25.
$\Rightarrow$ Probability $= \dfrac{{25}}{{30}}$
Cancel the same factor from numerator and denominator.
$\Rightarrow$Probability $= \dfrac{5}{6}$

Note: Students might make mistakes while calculating the numbers that are not divisible by 3 if they check each number from 1 to 30 whether it is divisible by 3 or not. It is easier to find the numbers that are divisible by 3 and then write that are not divisible by 3. Similar is with the part c where if you attempt to check each number it will take a lot of time and calculations, so we find the perfect squares lying between 1 and 30 and then subtract from the total number of cards.