Probability Example: Drawing Cards from a Deck

In this lecture, we discussed a probability problem involving drawing a card from a standard deck of 52 cards. The main focus was on calculating certain probabilities related to drawing red cards and kings. Let's break down the problem and solutions:

Key Events Defined

• R: Event of drawing a red card.
• K: Event of drawing a king.

Calculations and Concepts

Total Number of Outcomes (n(S))

• The sample space ( S ) is the total set of possible outcomes.
• For a standard deck:
  • Total number of outcomes (n(S)): 52

Number of Ways to Draw a Red Card (n(R))

• Red cards are in two suits (hearts and diamonds), each with 13 cards.
• Number of red cards (n(R)): 2 × 13 = 26

Probability of Drawing a Red Card

• Formula: ( P(R) = \frac{n(R)}{n(S)} )
• Calculation: ( P(R) = \frac{26}{52} = 0.5 )
  • There is a 50% chance of drawing a red card.

Number of Ways to Draw a King (n(K))

• There are four kings in a deck.
• Number of ways to draw a king (n(K)): 4

Probability of Drawing a King

• Formula: ( P(K) = \frac{n(K)}{n(S)} )
• Calculation: ( P(K) = \frac{4}{52} = 0.07692308 )
  • There is approximately a 7.7% chance of drawing a king.

Key Takeaways

• Sample Space (n(S)) is the total possible outcomes, which is 52 for a deck of cards.
• Counting Outcomes for specific events like red cards or kings helps determine probabilities.
• Probability Calculation involves dividing the number of favorable outcomes by the total number of outcomes.
• Interpretation: The results align with expectations; for example, red cards make up half the deck, while kings are relatively rare.

By understanding and applying these concepts, we can effectively determine probabilities for various events when dealing with a standard deck of cards.