Problem 15 - Thinking Mathematically

Question

In Exercises 15-20, you draw one card from a 52-card deck. Then the card is replaced in the deck, the deck is shuffled, and you draw again. Find the probability of drawing a picture card the first time and a heart the second time.

Step-by-Step Solution

Answer

The probability of drawing a picture card the first time and a heart the second time is \(\frac{3}{52}\).

Step 1: Calculate Probability of Drawing a Picture Card

First, determine the probability of drawing a picture card. There are 12 picture cards in a deck of 52. So the probability of drawing a picture card, \( P(Picture) \), is \( \frac{12}{52} \) or \( \frac{3}{13} \).

Step 2: Calculate Probability of Drawing a Heart Card

Second, determine the probability of drawing a heart. There are 13 heart cards in a deck of 52. So the probability of drawing a heart, \( P(Heart) \), is \( \frac{13}{52} \) or \( \frac{1}{4} \).

Step 3: Multiply Individual Probabilities

Finally, because the two events are independent, multiply these two probabilities to find the overall probability, \( P(Picture and Heart) = P(Picture) \times P(Heart) = \frac{3}{13} \times \frac{1}{4} = \frac{3}{52} \).

Key Concepts

Independent Events

When playing card games, the concept of independent events is crucial to understanding how the outcome of one event does not affect the outcome of another. In our textbook exercise scenario, drawing a card, replacing it in the deck, and then drawing again involves two separate events: drawing the picture card and drawing a heart. These are independent because the outcome of the first draw does not influence the second draw, as the card is replaced and the deck is shuffled.

When events are independent, their probabilities can simply be multiplied to find the combined probability of both events occurring in sequence. This is important because it simplifies the process of calculating complex probabilities in many card games, where events typically occur consecutively and independently.

Probability Calculation

Probability calculation is all about determining the likelihood of an event occurring. In our card game example, to calculate the probability of drawing particular cards, we divide the number of favorable outcomes by the total number of possible outcomes. For instance, the number of picture cards in a standard deck of 52 cards gives us the probability of drawing that picture card. Likewise, the probability of drawing a heart is calculated by the number of hearts in the deck.

Remember, the key to an accurate probability calculation is to ensure that the number of possibilities is current and correct, accounting for events that may change the likelihood of certain outcomes, such as cards being removed or replaced in the deck.

Combinatorics

Combinatorics plays a fundamental role in calculating probabilities in card games, as it involves the study of counting, combination, and permutation of sets of elements (in this case, cards). It helps us understand the different ways cards can be arranged and combined.

In our example, combinatory principles are simpler, since each event is treated independently due to the replacement and reshuffling of the cards. However, in more complex card game scenarios that do not involve replacement, such as poker or bridge, players must understand combinations and permutations to assess their chances of achieving a winning hand. Indeed, mastering combinatory techniques allows card players to make strategic decisions based on the likelihood of obtaining certain card combinations.