Question

All the cards in an ordinary deck of 52 cards are numbered from 1 to \(52 .\) If a card is drawn at random from the deck, then what is the probability that it will have a prime number? (1) \(\frac{7}{26}\) (2) \(\frac{15}{52}\) (3) \(\frac{5}{17}\) (4) \(\frac{4}{13}\)

Short answer

Answer: The probability of drawing a card with a prime number from a deck of 52 cards is \(\frac{15}{52}\).

Step-by-step solution

List the prime numbers between 1 and 52

Prime numbers are numbers that only have two factors: 1 and itself. List down all the prime numbers between 1 and 52. The prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

Count the number of prime numbers

Count the prime numbers that we listed in step 1: there are a total of 15 prime numbers between 1 and 52.

Calculate the probability

To calculate the probability, we need to divide the number of prime numbers (favorable outcomes) by the total number of cards in the deck (possible outcomes). So, the probability of drawing a card with a prime number is: \(\frac{15}{52}\)

Match the result with the options

Compare the probability we calculated in step 3, \(\frac{15}{52}\), with the given options. Our result matches with option 2. The correct answer is: (2) \(\frac{15}{52}\).

Key concepts

Understanding Prime Numbers

Prime numbers play a crucial role in various aspects of mathematics and its applications. A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In other words, it has exactly two distinct positive divisors: 1 and the number itself.

For example, the first six prime numbers are 2, 3, 5, 7, 11, and 13. As you can see, 2 is the only even prime number because any other even number can be divided by 2, which would then have at least three divisors. Understanding prime numbers is fundamental when solving problems related to the divisibility of numbers, cryptography, and prime factorization, among other things.

Prime numbers have a certain unpredictability and do not follow a specific pattern, making them an interesting and mysterious area of study in mathematics. Various algorithms and tests have been designed to identify prime numbers, but there is no formula to predict them.

The Basics of Probability Calculation

Probability is a branch of mathematics that deals with calculating the likelihood of various possible outcomes in an experiment or event. The probability of an event is a number between 0 and 1, where 0 indicates impossibility, and 1 indicates certainty. The fundamental formula for probability is:
\[\begin{equation}\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\end{equation}\]

To grasp the concept of probability, one must be familiar with terms such as 'favorable outcomes,' which refer to the number of ways an event can occur, and 'possible outcomes,' which are all the possible results of an event, regardless of whether they are favorable or not.

Hence, probability calculations are essentially a ratio of the favorable to all possible outcomes, which explains why they can also be expressed as fractions, decimals, or percentages. Effective probability calculation requires both an understanding of the event in question and the ability to count or identify the correct number of outcomes.

Solving Card Probability Problems

Card probability problems are a classic example of practical applications of probability in everyday life. A standard deck of cards has 52 cards, which are usually divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, numbered from 1 (Ace) to 10, and including three face cards — Jack, Queen, and King.

When dealing with card probability, it is essential to understand the composition of the deck and whether any conditions, such as specific numbers or suits, are being considered for the event. For instance, if the event is 'drawing a card with a prime number', you would first list all the prime numbers that fall within the range of the card numbers, and then calculate the probability based on these favorable outcomes compared to the total number of cards.

An important tip for solving card probability problems is to approach them methodically: note the total number of cards, determine the favorable outcomes based on the event, and then apply the basic probability formula. Such structured approaches greatly enhance the chances of correctly solving card probability problems.