Problem 19 - College Algebra

Question

A fair coin is tossed two times in succession. The sample space of equally- likely outcomes is \(\\{H H, H T, T H, T T\\} .\) Find the probability of getting: the same outcome on each toss.

Step-by-Step Solution

Answer

The probability of getting the same outcome on each toss is 0.5.

Step 1: Identify the sample space

We begin by identifying the sample space of equally likely outcomes. The sample space when a fair coin is tossed twice is given as \(\{H H, H T, T H, T T\}\). This means that there are four equally likely outcomes: getting heads on both tosses (H H), getting a head on the first toss and tail on the second (H T), getting a tail on the first toss and head on the second (T H), and finally getting tails on both tosses (T T).

Step 2: Identify the favorable outcomes

Next, we identify the favorable outcomes. A favorable outcome is a scenario which we are interested in finding the probability of. In this case, the exercise asks the probability of getting the same outcome on each toss. This is the case for two of our potential outcomes: 'H H' and 'T T'.

Step 3: Calculate the probability

To calculate the probability, we use the formula: Probability (P) = Number of favorable outcomes / Total number of outcomes. Here, the total number of outcomes is 4 (H H, H T, T H, T T), and the number of favorable outcomes is 2 (H H, T T). Plugging these values into the formula gives us: \(P = 2 / 4 = 0.5\).

Key Concepts

Sample Space

When discussing probability, one of the first concepts to understand is the sample space. The sample space is the set of all possible outcomes of a probability experiment. In the case of tossing a fair coin twice, the sample space can be represented as

• 'H H' – Heads on both tosses
• 'H T' – Head on the first toss, Tail on the second
• 'T H' – Tail on the first toss, Head on the second
• 'T T' – Tails on both tosses

The sample space helps us visualize all the outcomes that we might encounter in the experiment. This list is essential as it forms the foundation upon which the probability calculations are built. Each outcome in the sample space is considered when determining the probability of a particular event occurring. Understanding the sample space is key to accurately determining probabilities for any event related to the experiment.

Equally Likely Outcomes

Equally likely outcomes are a vital concept when understanding probability. They refer to outcomes that have the same chance of occurring. In probability theory, it's essential that each outcome in an experiment's sample space is considered equally likely unless stated otherwise.

For our coin toss example, the four outcomes ('H H', 'H T', 'T H', 'T T') are equally likely. Each has an equal probability of \[\frac{1}{4} \]of occurring. Recognizing that outcomes are equally likely is crucial because it allows us to fairly calculate probabilities. By knowing that each outcome is equally probable, you can apply mathematical formulas, like dividing the number of favorable outcomes by the total number of outcomes, to compute the probability of an event accurately. This concept ensures fairness and correctness in probabalistic calculations.

Favorable Outcomes

Favorable outcomes represent the specific outcomes of interest within a probability problem. These are the outcomes that align with the conditions of the event we wish to analyze. In our example of tossing a coin twice, we are interested in the probability of getting the same result on each toss.

The favorable outcomes for this event are 'H H' and 'T T', as both offer consistent results across the two tosses. Once identified, we can use the number of favorable outcomes to determine the probability of the event.

• Total number of favorable outcomes: 2 ('H H', 'T T')
• Total number of possible outcomes: 4
• Probability: \[\frac{2}{4} = 0.5\]

Comprehending what constitutes a favorable outcome allows us to precisely analyze the likelihood of various events. Accurately identifying these outcomes is essential in deriving meaningful probability calculations.