Problem 2 - Algebra for College Students

Question

Find the probability of tossing each of the following events: Two tails

Step-by-Step Solution

Answer

The probability of tossing two tails is 0.25.

Step 1: Determine the Sample Space

When tossing a fair coin twice, the possible outcomes are HH, HT, TH, and TT. Therefore, the sample space consists of 4 equally likely outcomes, each representing the result of the two coin tosses.

Step 2: Identify the Specific Event

The event of interest is obtaining two tails (TT) when tossing the coins. In the sample space {HH, HT, TH, TT}, only 'TT' satisfies the condition of this event.

Step 3: Calculate the Probability of the Event

Since each outcome in the sample space is equally likely, the probability of any specific event is given by the number of successful outcomes over the total number of possible outcomes. Here, there is 1 successful outcome (TT) out of 4 possible outcomes. Hence, the probability is calculated as:\[P(\text{two tails}) = \frac{1}{4} = 0.25\]

Key Concepts

Sample Space

Understanding the concept of sample space is crucial when dealing with probability problems. It represents all possible outcomes of a random experiment.
In the context of coin tosses, a sample space includes all potential sequences of outcomes.
For example, when tossing two coins, each can either land on heads (H) or tails (T).

• Possible outcomes: Each coin can independently show H or T.
• Sample Space: For two tosses, the sample space is {HH, HT, TH, TT}, meaning there are four possible outcomes.

This sample space helps organize our thinking about the problem and is essential for calculating probabilities.

Coin Tosses

A coin toss is a fundamental experiment in probability, often used to illustrate basic concepts. It is simple yet powerful, as each toss can result in one of two outcomes.
These basic units are heads (H) or tails (T).
When two coins are tossed, or one coin is tossed twice, the sequence of outcomes forms a more complex sample space. In our exercise, we are analyzing pairs of results:

• First toss can be H or T.
• Second toss can be H or T.
• These combinations create the sample space: {HH, HT, TH, TT}.

Coin toss problems help provide a clear understanding of how probabilities add up. It's especially useful for visualizing and calculating the likelihood of multiple sequential events.

Fair Coin

A fair coin is one that has no bias, meaning it is equally likely to land on heads or tails. This fairness is crucial for modeling accurate probabilities.
When a coin is fair, each side has a probability of \[P(H) = \frac{1}{2} \quad \text{and} \quad P(T) = \frac{1}{2}\]

• This property ensures each side appears equally over many tosses.
• Such fairness is assumed in theoretical exercises, like the one discussed here.

This assumption keeps calculations straightforward, letting us focus on the structure of more complex probability problems without worrying about biases.

Outcomes

Outcomes are the results that arise from performing a random experiment. For coin tosses, these outcomes are based on the positioning of heads or tails after each toss.
In a two-coin toss scenario, where both can land either H or T, possible outcomes become the cornerstone of probability calculations.

• Each unique sequence of heads and tails forms an outcome, such as HH, HT, TH, TT.
• In probability exercises, identifying specific desired outcomes is vital. For example, wanting two tails results in the outcome TT.

Understanding outcomes makes it clear which events are possible, aiding in the efficient calculation of probabilities by relating desired events to all possible results.