Problem 22 - Algebra and Trigonometry

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 $$\text{the same outcome on each toss.}$$

Step-by-Step Solution

Answer

The probability of getting the same outcome on each toss is \(\frac{1}{2}\) or 0.5.

Step 1 - Identify Relevant Outcomes

First, identify the outcomes that involve the same result on both tosses. Out of the sample space \(\{H H, H T, T H, T T\}\), these are \(H H\) and \(T T\).

Step 2 - Determine Total And Favorable Outcomes

The total number of outcomes is the number of possible results when the coin is tossed twice, which amount to 4. The favorable outcomes, that is, those that satisfy our condition, are 2 (Two Heads or Two Tails).

Step 3 - Calculate the Probability

The probability of an event is calculated as the number of favorable outcomes divided by the total number of outcomes. Thus, the probability of getting the same result on both tosses is \(\frac{2}{4}\).

Key Concepts

Sample Space

In probability theory, the term 'sample space' refers to the set of all possible outcomes of an experiment. In the given exercise, a fair coin is tossed twice, which makes the experiment's sample space quite manageable. For this example, the sample space is defined as a collection of all potential sequences of heads (H) and tails (T) for the two tosses.
Thus, the sample space is:

• HH (heads on both tosses)
• HT (heads first, then tails)
• TH (tails first, then heads)
• TT (tails on both tosses)

Each sequence counts as a distinct and separate outcome, and all are considered when computing probabilities. Hence, it ensures no possibilities are overlooked when solving problems related to the event.

Favorable Outcomes

'Favorable outcomes' in a probability context are the outcomes of a sample space that meet specified conditions of an event we are analyzing. In the original exercise, the condition of interest is getting the same result on both coin tosses.
From our sample space

• HH
• HT
• TH
• TT

Only 'HH' and 'TT' satisfy the condition of being the same on both tosses. Therefore, there are 2 favorable outcomes here. This concrete identification of what constitutes 'favorable' is crucial, as it directly impacts the calculation of probability.

Equally Likely Outcomes

The concept of 'equally likely outcomes' is fundamental in probability. It indicates that each result in the sample space has the same chance of occurring. When tossing a fair coin, every potential outcome is equally probable because the coin is unbiased.
In our case, each of the four outcomes in the sample space - HH, HT, TH, and TT - has an equal likelihood of appearing. This assumption simplifies the probability calculations significantly. For instance, when determining the probability for an event, we simply count how many outcomes meet the criteria and divide by the total number of equally likely outcomes to arrive at the probability value.