Question

Your friend has two standard decks of 52 playing cards and asks you to randomly draw one card from each deck. What is the probability that you will draw two spades.

Short answer

The probability of drawing two spades, one from each deck, is 1/16.

Step-by-step solution

Identify Total Number of Outcomes

In a standard deck of 52 playing cards, there are 4 suits - hearts, diamonds, clubs, and spades - each comprising 13 cards. Thus, the total number of outcomes, when a card is drawn from one deck, is 52.

Identify Favorable Outcomes

Since the task is to draw a spade, the favorable outcomes would correspond to the number of spades in one deck. There are 13 spades in a single deck.

Calculate Probability for One Event

The probability of drawing a spade from one deck, denoted by P(S1), can be calculated using the formula for the probability: the number of favorable outcomes divided by the total number of outcomes. Here, P(S1) = 13 / 52 = 1 / 4.

Calculate Overall Probability

Since the card drawing events are independent (drawing a card from one deck does not affect the probability of drawing a card from the other deck), the overall probability of drawing two spades, one from each deck, is given by the product of the individual probabilities. This can be calculated as: P(S1 and S2) = P(S1) * P(S2) = 1/4 * 1/4 = 1/16.

Key concepts

Independent Events

In probability, independent events are those whose outcomes do not influence each other. For example, consider drawing a card from two separate decks. The result of the card drawn from the first deck does not affect what card is drawn from the second deck. This is because each deck operates independently of the other.

• Independent events simplify probability calculations because the probability of both events is simply the product of their individual probabilities.
• This allows us to assess complex scenarios by breaking them into simpler, manageable parts.

Understanding these properties can help you evaluate probabilities more effectively in real-world and theoretical situations.

Favorable Outcomes

Favorable outcomes are the results that match the conditions we are looking for. In the context of our card-drawing exercise, favorable outcomes would be drawing a spade from a deck of cards.

• From a standard deck of cards, there are 13 spades, making 13 potential favorable outcomes.
• Identifying favorable outcomes is crucial for calculating probability, as they form the numerator in the probability formula.

By clearly defining what a favorable outcome is, you can more easily apply probability concepts across various scenarios.

Probability Calculation

Calculating probability involves several steps that allow us to quantify how likely an event is to occur. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Example

- For one deck, the probability of drawing a spade is computed as 13 favorable outcomes (spades) divided by the total 52 outcomes (entire deck), resulting in \( P(\text{spade}) = \frac{13}{52} = \frac{1}{4} \).
- To find the probability of drawing two spades, one from each of the two independent decks, you multiply these probabilities: \( P(\text{spade from both decks}) = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \).
By mastering probability calculations, you can advance in not only card games but in diverse areas such as statistics, science, and investment strategies.

Card Deck Probability

Calculating probabilities with a card deck is a classic way to understand basic probability concepts. A standard deck consists of 52 cards divided into four suits: hearts, diamonds, clubs, and spades.

• Each suit contains 13 cards, resulting in an equal probability of drawing a specific suit from the deck.
• If you're focused on one suit, like spades, understanding that there are 13 spades helps determine the likelihood of drawing one.

Card deck probability provides a tangible way to engage with abstract probability concepts, making it easier to visualize and understand probability in other contexts.