Problem 44 - Algebra and Trigonometry

Question

You are dealt one card from a 52-card deck. Find the probability that you are dealt a 5 or a black card.

Step-by-Step Solution

Answer

The probability that you are dealt a 5 or a black card is \(\frac{5}{13}\)

Step 1: Calculate the probability of being dealt a 5

There are 4 fives in a 52-card deck. So the probability is calculated by dividing the number of favourable outcomes by the total number of outcomes, which gives us \(\frac{4}{52} = \frac{1}{13}\).

Step 2: Calculate the probability of being dealt a black card

There are 26 black cards in a 52-card deck. So the probability is \(\frac{26}{52} = \frac{1}{2}\).

Step 3: Calculate the overlap

The overlap here would be drawing a black 5. There are 2 black fives in a 52 card deck, so the probability is \(\frac{2}{52} = \frac{1}{26}\).

Step 4: Combine the probabilities

We can't just add the probabilities we calculated in steps 1 and 2 because that would count the overlap twice. So we should add them together and then subtract the overlap, which gives us \(\frac{1}{13} + \frac{1}{2} - \frac{1}{26} = \frac{10}{26} = \frac{5}{13}\) .

Key Concepts

Probability of events

Probability is a measure of how likely an event is to occur. When we talk about the probability of an event, we express this likelihood as a fraction. The fraction consists of the number of favorable outcomes divided by the total number of possible outcomes.
This is the foundation of probability calculations. In the context of cards, it's important to know that a standard deck has 52 cards, allowing us to determine probabilities quickly.
For example, if you want to find the probability of drawing a 5 from a deck, you determine there are 4 such cards (since there are 4 suits in a deck, each with a 5) which leads to a probability calculation of \( \frac{4}{52} \), simplifying to \( \frac{1}{13} \).
Understanding this basic concept is crucial for working with more complex probability scenarios.

Overlapping events

When dealing with probability, overlapping events are scenarios where two events can happen at the same time. This is common in card games.
Let's say you're interested in finding the probability of drawing a 5 OR a black card from a deck. Both events have their own individual probabilities, but these events overlap.
The overlap occurs with cards that are both fives and black, such as the 5 of Spades and the 5 of Clubs. There are 2 such overlapping cards.
Ignoring overlaps can lead to incorrect probability calculations, as overlaps might be counted multiple times. In this exercise, we calculated the overlap as \( \frac{2}{52} = \frac{1}{26} \). You have to subtract this overlap from the sum of individual probabilities to get the correct answer.
Remember that when combining probabilities of events that can occur together, always adjust for any overlap.

Deck of cards

A deck of cards is a common example used to illustrate basic probability concepts. Understanding the composition of a deck is key to calculating probabilities.
A standard deck contains 52 cards divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards, from ace to king. The red suits are hearts and diamonds, while the black suits are clubs and spades.
Knowing the deck's structure helps you quickly determine the number of certain types of cards, such as black cards or face cards.
For example, for the exercise problem, knowing there are 26 black cards means the probability of drawing one is \( \frac{26}{52} = \frac{1}{2} \).
Mastering how the deck is arranged ensures you can apply probability calculations with confidence and accuracy.