A Box Contains Cards Numbered From 3 To 102. If One Card Is Selected Randomly From The Box, Find The Probability That It Bears:(i) A Perfect Square (ii) A Multiple Of 3 (iii) A Prime Number (iv) A Composite Number

by ADMIN 217 views

Introduction

Probability and number theory are two fundamental branches of mathematics that have numerous real-world applications. In this article, we will delve into a box containing cards numbered from 3 to 102 and explore the probability of selecting a card that bears specific properties. We will examine the probability of selecting a perfect square, a multiple of 3, a prime number, and a composite number.

Perfect Squares

A perfect square is a number that can be expressed as the square of an integer. In the given range, the perfect squares are 9, 16, 25, 36, 49, 64, 81, and 100. There are 8 perfect squares in the range from 3 to 102.

Calculating the Probability of Selecting a Perfect Square

To calculate the probability of selecting a perfect square, we need to divide the number of perfect squares by the total number of cards in the box.

import math

start = 3
end = 102



total_cards = end - start + 1



perfect_squares = [9, 16, 25, 36, 49, 64, 81, 100]



num_perfect_squares = len(perfect_squares)



probability_perfect_square = num_perfect_squares / total_cards


print("The probability of selecting a perfect square is:", probability_perfect_square)

Multiples of 3

A multiple of 3 is a number that can be expressed as 3 times an integer. In the given range, the multiples of 3 are 3, 6, 9, 12, ..., 99, 102. We can use the formula for the nth term of an arithmetic progression to find the number of multiples of 3.

# Define the first and last terms of the arithmetic progression
first_term = 3
last_term = 102


common_difference = 3



num_terms = (last_term - first_term) / common_difference + 1


print("The number of multiples of 3 is:", num_terms)

Calculating the Probability of Selecting a Multiple of 3

To calculate the probability of selecting a multiple of 3, we need to divide the number of multiples of 3 by the total number of cards in the box.

# Calculate the probability of selecting a multiple of 3
probability_multiple_of_3 = num_terms / total_cards

print("The probability of selecting a multiple of 3 is:", probability_multiple_of_3)

Prime Numbers

A prime number is a number that is divisible only by itself and 1. In the given range, the prime numbers are 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, and 101.

Calculating the Probability of Selecting a Prime Number

To calculate the probability of selecting a prime number, we need to divide the number of prime numbers by the total number of cards in the box.

# Define the prime numbers in the range
prime_numbers = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101]


num_prime_numbers = len(prime_numbers)



probability_prime_number = num_prime_numbers / total_cards


print("The probability of selecting a prime number is:", probability_prime_number)

Composite Numbers

A composite number is a number that is not prime. In the given range, the composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, and 102.

Calculating the Probability of Selecting a Composite Number

To calculate the probability of selecting a composite number, we need to divide the number of composite numbers by the total number of cards in the box.

# Define the composite numbers in the range
composite_numbers = [4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102]


num_composite_numbers = len(composite_numbers)



probability_composite_number = num_composite_numbers / total_cards


print("The probability of selecting a composite number is:", probability_composite_number)

Conclusion

Introduction

In our previous article, we explored the probability of selecting a card from a box containing numbers from 3 to 102 that bears specific properties. We calculated the probability of selecting a perfect square, a multiple of 3, a prime number, and a composite number. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the probability of selecting a number that is not a multiple of 3?

A: To calculate the probability of selecting a number that is not a multiple of 3, we need to subtract the probability of selecting a multiple of 3 from 1.

# Calculate the probability of selecting a number that is not a multiple of 3
probability_not_multiple_of_3 = 1 - probability_multiple_of_3

print("The probability of selecting a number that is not a multiple of 3 is:", probability_not_multiple_of_3)

Q: What is the probability of selecting a prime number that is greater than 10?

A: To calculate the probability of selecting a prime number that is greater than 10, we need to divide the number of prime numbers greater than 10 by the total number of cards in the box.

# Define the prime numbers greater than 10
prime_numbers_greater_than_10 = [11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101]


num_prime_numbers_greater_than_10 = len(prime_numbers_greater_than_10)



probability_prime_number_greater_than_10 = num_prime_numbers_greater_than_10 / total_cards


print("The probability of selecting a prime number that is greater than 10 is:", probability_prime_number_greater_than_10)

Q: What is the probability of selecting a composite number that is greater than 50?

A: To calculate the probability of selecting a composite number that is greater than 50, we need to divide the number of composite numbers greater than 50 by the total number of cards in the box.

# Define the composite numbers greater than 50
composite_numbers_greater_than_50 = [52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102]


num_composite_numbers_greater_than_50 = len(composite_numbers_greater_than_50)



probability_composite_number_greater_than_50 = num_composite_numbers_greater_than_50 / total_cards


print("The probability of selecting a composite number that is greater than 50 is:", probability_composite_number_greater_than_50)

Q: What is the probability of selecting a number that is a perfect square and a multiple of 3?

A: To calculate the probability of selecting a number that is a perfect square and a multiple of 3, we need to divide the number of perfect squares that are multiples of 3 by the total number of cards in the box.

# Define the perfect squares that are multiples of 3
perfect_squares_multiple_of_3 = [9, 36, 81]


num_perfect_squares_multiple_of_3 = len(perfect_squares_multiple_of_3)



probability_perfect_square_multiple_of_3 = num_perfect_squares_multiple_of_3 / total_cards


print("The probability of selecting a number that is a perfect square and a multiple of 3 is:", probability_perfect_square_multiple_of_3)

Conclusion

In this article, we answered some frequently asked questions related to the topic of probability and number theory. We calculated the probability of selecting a number that is not a multiple of 3, a prime number that is greater than 10, a composite number that is greater than 50, and a number that is a perfect square and a multiple of 3. These results demonstrate the importance of understanding probability and number theory in real-world applications.