Problem 2 - Thinking Mathematically

Question

You are dealt one card from a 52-card deck. Find the probability that you are not dealt a \(3 .\)

Step-by-Step Solution

Answer

The probability that one is not dealt a 3 from a 52-card deck is \( \frac{12}{13} \)

Step 1: Identify total and desired outcomes

The total outcomes when one card is dealt from a 52-card deck is 52. However, since being dealt a 3 isn't the desired outcome, the desired outcomes will exclude four 3s (3 of hearts, 3 of diamonds, 3 of clubs, 3 of spades). This gives us \( 52 - 4 = 48 \) desired outcomes.

Step 2: Calculating Probability

The formula to calculate probability is given by desired outcomes divided by total outcomes. Substituting the numbers calculated into the formula, we get the probability as \( \frac{48}{52} \)

Step 3: Simplifying the Fraction

Simplify the fraction \( \frac{48}{52} \) to its lowest terms. We can do this by dividing both the numerator and the denominator by the greatest common divisor, which is 4 in this case. This results in \( \frac{48 \div 4}{52 \div 4} = \frac{12}{13} \)

Key Concepts

Card Deck

A standard card deck is composed of 52 individual cards. These are divided into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, numbered from 2 to 10, including three face cards: Jack, Queen, and King, plus the Ace. All these add up to make a complete deck.
Understanding the structure of a card deck is essential when calculating probabilities because it helps you know the total number of possible outcomes. When you are asked about the probability of drawing a specific type of card, like not getting a particular number, you start with these 52 options.
To calculate probabilities effectively, this foundational understanding of a card deck is crucial.

Desired Outcomes

In any probability problem, understanding what constitutes a 'desired outcome' is vital. Desired outcomes are those results that meet the criteria of what you are trying to find out.
In this problem, the aim is to figure out the likelihood of not being dealt a '3'. You have to exclude all the '3' cards from your outcomes. Since there are four '3's in the deck (one for each suit), you subtract those from the total.

• Total cards: 52
• Cards that are '3': 4 (one in each suit)
• Desired outcomes: 52 - 4 = 48

This calculation gives you the total number of successful outcomes, which means drawing any other card apart from a '3'.

Calculating Probability

Probability helps you understand the chance of a specific event happening. It's calculated by dividing the number of desired outcomes by the total number of possible outcomes.
Using the problem given, the total outcomes are all 52 cards, because you could draw any card. The desired outcomes, as calculated before, are 48.

Hence, the probability formula is:
\[\text{Probability} = \frac{\text{Number of Desired Outcomes}}{\text{Total Outcomes}} = \frac{48}{52}\]
This fraction represents the likelihood of drawing a card that is not a '3' from a standard deck.

Simplifying Fractions

A key part of calculating probability often involves simplifying fractions. Simplification helps make your answer easier to understand at a glance.
Here, the fraction obtained was \( \frac{48}{52} \). To simplify, find the greatest common divisor (GCD) of the numerator and the denominator. In this case, GCD is 4. You divide both 48 and 52 by this number:

• \(\frac{48 \div 4}{52 \div 4} = \frac{12}{13}\)

Simplifying does not change the value of the probability; it simply makes it more neat and straightforward. The simplified form, \( \frac{12}{13}\), expresses the same likelihood of drawing a non-'3' card.