Problem 6 - Algebra and Trigonometry

Question

A card is drawn randomly from a standard 52-card deck. Find the probability of the given event. (a) The card drawn is a heart. (b) The card drawn is either a heart or a spade. (c) The card drawn is a heart, a diamond, or a spade.

Step-by-Step Solution

Answer

(a) 1/4, (b) 1/2, (c) 3/4.

Step 1: Understand the Structure of a Deck

A standard deck of cards contains 52 cards. These are divided into four suits: hearts, spades, diamonds, and clubs. Each suit has 13 cards.

Step 2: Calculate Probability of a Heart

To find the probability of drawing a heart, note that there are 13 hearts in the deck. The probability is the ratio of favorable outcomes (hearts) to the total number of cards. Thus, the probability is given by: \[ P( ext{heart}) = \frac{13}{52} \] Simplifying gives: \[ P( ext{heart}) = \frac{1}{4} \]

Step 3: Calculate Probability of a Heart or a Spade

Both hearts and spades have 13 cards each. Thus, there are a total of 26 favorable outcomes for this event. The probability of drawing either a heart or a spade is: \[ P( ext{heart or spade}) = \frac{26}{52} \] Simplifying gives: \[ P( ext{heart or spade}) = \frac{1}{2} \]

Step 4: Calculate Probability of a Heart, Diamond, or Spade

Hearts, diamonds, and spades each have 13 cards. Adding these gives us 39 cards. Therefore, the probability of drawing one of these is: \[ P( ext{heart, diamond, or spade}) = \frac{39}{52} \] Simplifying gives: \[ P( ext{heart, diamond, or spade}) = \frac{3}{4} \]

Key Concepts

Standard Deck of Cards

Understanding a standard deck of cards helps when calculating probabilities. A standard deck consists of 52 playing cards, which are divided into four distinct suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, ranging from Ace through to King.
It is important to note that there are two red suits - hearts and diamonds, and two black suits - clubs and spades. This uniform distribution of suits and values within the deck forms the basis for various probability calculations.

Favorable Outcomes

In the realm of probability, a single outcome or a group of outcomes that satisfy an event is considered 'favorable'. For example, if you're event is drawing a heart, all the heart cards are considered favorable outcomes.
To clearly determine the favorable outcomes of a specific event:

• Identify the criteria of the event (e.g. drawing a heart).
• Count the number of outcomes in the deck that meet this criteria. For hearts, this is 13.
• In case multiple events are grouped, add the total count of each individual event (e.g. hearts and spades are 26 favorable outcomes).

Favorable outcomes are the backbone for calculating probability as they specifically define the conditions we are assessing.

Probability Calculation

Probability is a measure of how likely it is for an event to occur. In using a deck of cards, probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
For a specific event like drawing a heart, probability can be calculated using the formula:
\[ P( ext{event}) = \frac{ ext{favorable outcomes}}{ ext{total outcomes}} \] For example, since there are 13 hearts in a deck of 52 cards, drawing a heart would be:
\[ P( ext{heart}) = \frac{13}{52} = \frac{1}{4} \]
When calculating probability for events with multiple outcomes, such as drawing a heart or a spade, simply sum the favorable outcomes for each category before dividing by the total outcomes:
\[ P( ext{heart or spade}) = \frac{26}{52} = \frac{1}{2} \] Probability calculations are fundamental in assessing risk and determining expectations.