Problem 10 - Algebra for College Students

Question

If a single coin is tossed twice, then what is the probability of getting a) heads followed by tails? b) two heads in a row? c) a tail on the second toss? d) exactly one tail?

Step-by-Step Solution

Answer

a) 0.25 b) 0.25 c) 0.50 d) 0.50

Step 1: Understand the basic outcomes

When a coin is tossed twice, there are four possible outcomes: HH (heads-heads), HT (heads-tails), TH (tails-heads), and TT (tails-tails). Each outcome has an equal probability of occurring.

Step 2: Calculate the probability of heads followed by tails

Out of the four possible outcomes, only HT represents heads followed by tails. Hence, the probability is the number of favorable outcomes divided by the total number of outcomes: \[ P(\text{HT}) = \frac{1}{4} \text{ or } 0.25 \text{ or } 25\text{\text{%.}} \]

Step 3: Calculate the probability of two heads in a row

Out of the four possible outcomes, only HH represents two heads in a row. Hence, the probability is: \[ P(\text{HH}) = \frac{1}{4} \text{ or } 0.25 \text{ or } 25\text{\text{%.}}\]

Step 4: Calculate the probability of a tail on the second toss

We need to count the number of outcomes where the second toss is tails, which are HT and TT. Hence, the probability is: \[ P(\text{tail on second toss}) = \frac{2}{4} \text{ or } \frac{1}{2} \text{ or } 50\text{\text{%.}}\]

Step 5: Calculate the probability of exactly one tail

The outcomes with exactly one tail are HT and TH. Hence, the probability is: \[ P(\text{exactly one tail}) = \frac{2}{4} \text{ or } \frac{1}{2} \text{ or } 50\text{\text{%.}}\]

Key Concepts

headline of the respective core concept

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event will never occur, and 1 means it is certain to occur. In everyday language, we often use percentages to express probability. For example, a probability of 0.75 or ¾ is 75%. Basic probability concepts are crucial to understanding more complex problems. So let's dive into these fundamental ideas!

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Outcome calculation involves identifying all possible scenarios that can happen in a given situation. For example, when you toss a coin twice, four outcomes are possible: \(HH\) (heads-heads), \(HT\) (heads-tails), \(TH\) (tails-heads), and \(TT\) (tails-tails). Each outcome has an equal chance of happening since a fair coin has no bias, making the probability of each individual outcome \(\frac{1}{4}\). Recognizing and listing all possible outcomes is a vital step before moving on to more specific probability questions. It ensures you do not miss any possible scenario, leading to accurate calculations.

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To find the probability of multiple events, such as getting heads followed by tails or two heads in a row, you use the ratio of favorable outcomes to the total number of outcomes.
For heads followed by tails, there is only one favorable outcome among the four possible ones, that is \(HT\). Therefore, the probability is \(\frac{1}{4}\).
Similarly, for two heads in a row, the only favorable outcome is \(HH\). Thus, the probability is also \(\frac{1}{4}\).
Understanding how to compute multiple events' probabilities is essential for dealing with more complicated probabilistic scenarios.

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Coin toss probabilities often serve as introductory examples to understand random events and probability calculations. In our exercise, we consider the following:

• The probability of getting a tail on the second toss: To find this, look at outcomes where the second coin shows tails (\(HT\) and \(TT\)). There are two favorable outcomes out of four, so the probability is \(\frac{2}{4}\) or 0.5 (50%).

• The probability of getting exactly one tail: This includes outcomes where there is one head and one tail (\(HT\) and \(TH\)). There are two favorable outcomes out of four, giving us a probability of \(\frac{2}{4}\) or 0.5 (50%).

Coin toss probabilities provide a clear and straightforward way to understand how these basic principles apply to random events.