A card is drawn from a pack of 52 cards. A gambler bets that it is either a spade or an ace. The odds against his winning are
9 ∶ 4
Understanding the Problem: Card Probability and Odds
This question asks us to calculate the odds against a specific event happening when a single card is drawn from a standard deck of 52 cards. The event is that the card drawn is either a spade or an ace. We need to determine the ratio of unfavorable outcomes to favorable outcomes for this event.
Calculating Favorable Outcomes (Winning)
The event we are interested in is drawing a card that is a spade or an ace. In a standard 52-card deck:
- There are 13 spade cards.
- There are 4 ace cards (Ace of Spades, Ace of Hearts, Ace of Diamonds, Ace of Clubs).
Notice that the Ace of Spades is counted in both categories. To find the total number of cards that are either a spade or an ace, we use the principle of inclusion-exclusion:
Number of (Spade or Ace) = Number of Spades + Number of Aces - Number of (Spade and Ace)
The card that is both a spade and an ace is the Ace of Spades. So, there is 1 such card.
Number of favorable outcomes = \(13 + 4 - 1 = 16\)
There are 16 cards in the deck that are either a spade or an ace.
Calculating Unfavorable Outcomes (Losing)
Unfavorable outcomes are drawing a card that is neither a spade nor an ace. The total number of cards in the deck is 52.
Number of unfavorable outcomes = Total cards - Number of favorable outcomes
Number of unfavorable outcomes = \(52 - 16 = 36\)
There are 36 cards in the deck that are neither a spade nor an ace.
Calculating Odds Against Winning
Odds against an event are defined as the ratio of the number of unfavorable outcomes to the number of favorable outcomes.
Odds Against Winning = Number of Unfavorable Outcomes : Number of Favorable Outcomes
Odds Against Winning = \(36 : 16\)
This ratio can be simplified by dividing both numbers by their greatest common divisor, which is 4.
\(36 \div 4 = 9\)
\(16 \div 4 = 4\)
So, the simplified odds against winning are \(9 : 4\).
Summary of Outcomes
| Type of Outcome | Number of Cards |
|---|---|
| Favorable (Spade or Ace) | 16 |
| Unfavorable (Neither Spade nor Ace) | 36 |
| Total Cards | 52 |
The odds against the gambler winning are 36 to 16, which simplifies to 9 to 4.
Revision Table: Key Probability Concepts
| Concept | Definition/Formula |
|---|---|
| Probability of an Event (E) | P(E) = (Number of favorable outcomes) / (Total number of outcomes) |
| Probability of Not E (E') | P(E') = 1 - P(E) or P(E') = (Number of unfavorable outcomes) / (Total number of outcomes) |
| Odds For an Event | P(E) : P(E') or (Favorable Outcomes) : (Unfavorable Outcomes) |
| Odds Against an Event | P(E') : P(E) or (Unfavorable Outcomes) : (Favorable Outcomes) |
| Inclusion-Exclusion Principle (for A or B) | P(A \(\cup\) B) = P(A) + P(B) - P(A \(\cap\) B) |
Additional Information: Standard Deck Structure
A standard deck of 52 cards consists of four suits:
- Spades (♠)
- Hearts (♥)
- Diamonds (♦)
- Clubs (♣)
Each suit has 13 cards:
- Ace (A)
- Numbered cards from 2 to 10
- Face cards: Jack (J), Queen (Q), King (K)
Understanding the composition of the deck is crucial for solving card probability problems.
Important Questions from Probability of Random Experiments
Two cards are drawn successively without replacement from a well-shuffled pack of 52 cards. The probability of drawing two aces is
A biased coin with the probability of getting head equal to \(\frac{1}{4}\) is tossed five times. What is the probability of getting tail in all the first four tosses followed by head ?
Three dice are thrown. What is the probability that each face shows only multiples of 3 ?
A natural number n is chosen from the first 50 natural numbers. What is the probability that \(n+\frac{50}{n}<50 \) ?
A card is drawn from a well-shuffled deck of 52 cards. What is the probability that it is queen of spade?
