Odds The chances of winning are often written in terms of odds rather than probabilities. The odds of winning is the ratio of the number of successful outcomes to the number of unsuccessful outcomes. The odds of losing is the ratio of the number of unsuccessful outcomes to the number of successful outcomes. For example, when the number of successful outcomes is 2 and the number of unsuccessful outcomes is 3, the odds of winning are 2 : 3 (read "2 to 3"). In Exercises 91-96, use this information about odds.
94. A card is picked at random from a standard deck of 52 playing cards. Find the odds that it is a spade.

Verified step by step guidance

1. Step 1: Understand the problem. A standard deck of 52 playing cards is divided into 4 suits: spades, hearts, diamonds, and clubs. Each suit contains 13 cards. We are tasked with finding the odds that a randomly picked card is a spade.
2. Step 2: Recall the formula for odds. The odds of an event occurring is the ratio of the number of successful outcomes to the number of unsuccessful outcomes. Mathematically, this can be expressed as: $\frac{\mathrm{Number\; of\; Successful\; Outcomes}}{\mathrm{Number\; of\; Unsuccessful\; Outcomes}}$.
3. Step 3: Identify the successful outcomes. Since there are 13 spades in the deck, the number of successful outcomes is 13.
4. Step 4: Identify the unsuccessful outcomes. The total number of cards in the deck is 52, and the number of unsuccessful outcomes is the total number of cards minus the number of spades: $52-13$.
5. Step 5: Write the odds. The odds of picking a spade are the ratio of successful outcomes (13) to unsuccessful outcomes (52 - 13). Simplify this ratio if possible to express the final odds in the form of 'a : b'.

Key Concepts

Odds vs. Probability

Odds represent the ratio of successful outcomes to unsuccessful outcomes, while probability measures the likelihood of an event occurring out of all possible outcomes. For instance, if the odds of winning are 2:3, the probability of winning is calculated as the number of successful outcomes divided by the total number of outcomes, which in this case would be 2 out of 5.

Standard Deck of Cards

A standard deck of playing cards consists of 52 cards, divided into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards. Understanding the composition of the deck is essential for calculating the odds of drawing a specific suit, such as spades, which directly influences the outcome of the problem presented.

Calculating Odds

To calculate the odds of an event, you need to determine the number of successful outcomes and the number of unsuccessful outcomes. For example, if there are 13 spades in a deck of 52 cards, the odds of drawing a spade would be 13 successful outcomes to 39 unsuccessful outcomes, resulting in odds of 13:39, which can be simplified to 1:3.