Problem 60 - Algebra for College Students

Question

What are the odds against getting four tails with a toss of four coins? 15 to 1

Step-by-Step Solution

Answer

The odds against getting four tails with a toss of four coins are 15 to 1.

Step 1: Understanding the Problem

We are tasked to find the odds against getting four tails when tossing four coins. This means we need to calculate the probability of four tails, and then find the odds against this event happening.

Step 2: Calculate Probability of Four Tails

The probability of a single coin landing on tails is 0.5. Since each coin is independent, the probability of all four coins landing on tails is \[ P( ext{four tails}) = (0.5)^4 = 0.0625. \]

Step 3: Calculate Probability of Not Getting Four Tails

To find the probability of not getting four tails, subtract the probability of getting four tails from 1:\[ P( ext{not four tails}) = 1 - 0.0625 = 0.9375. \]

Step 4: Calculate Odds Against Four Tails

Odds against an event is given by the ratio of the probability of the event not happening to the probability of the event happening. Thus,\[ \text{Odds against four tails} = \frac{0.9375}{0.0625} = 15. \] This means the odds are 15 to 1 against getting four tails.

Key Concepts

Independent Events

In probability, an event is independent if the occurrence of one event does not affect the occurrence of another. When you toss a coin, the outcome of one toss does not influence the outcomes of subsequent tosses. Each flip of a coin is an example of an independent event because the result of one flip is not dependent on the previous or next flip. Understanding independent events is crucial for calculating probabilities in experiments that involve random processes.

• Independence implies that the probability of multiple independent events occurring together is the product of their individual probabilities.
• This is why when calculating the probability of getting four tails in a row from tossing four coins, you multiply the probability of a single tail (0.5) four times: \[(0.5) \times (0.5) \times (0.5) \times (0.5) = (0.5)^4.\]

Odds Calculation

Odds offer a different way to express probabilities and provide insights into the likelihood of an event. To understand odds, remember:

• Odds in favor: This is the ratio of the probability of an event happening to the probability of it not happening.
• Odds against: Conversely, this is the ratio of the probability of an event not happening to the probability of it happening.

When calculating odds against an event, like in the original exercise, you need to determine both the probability of the event occurring and the probability of it not occurring.- With our coin toss scenario, the odds against getting four tails was calculated as:\[\text{Odds} = \frac{\text{Probability of not getting four tails}}{\text{Probability of getting four tails}} = \frac{0.9375}{0.0625} = 15.\]This result means that for every 15 times you do not get four tails, there is 1 time you do.

Coin Toss Experiment

The coin toss experiment is one of the simplest yet powerful demonstrations of probability. It involves flipping a fair coin, which has two equally likely outcomes: heads or tails.

• Each coin toss is independent, making it an excellent tool to illustrate basic probability concepts.
• In an experiment involving multiple coin tosses, like tossing four coins at once, calculating the probability of getting a specific combination, such as all tails, can be fascinating.

For a single coin, the probability of landing on tails is 0.5. When tossing four coins simultaneously, because each toss is independent, you multiply the probability of a tail from each toss: \[(0.5)^4 = 0.0625.\]This means there is a 6.25% chance of getting four tails in four tosses, reflecting the power and simplicity of probability in coin toss experiments.