Question

Suppose three fair coins are tossed. What is the probability that precisely two coins land heads up if the first coin lands heads up and the second coin lands tails up?

Short answer

The probability is \( \frac{1}{2} \).

Step-by-step solution

Understand the Problem

We need to find the probability that precisely two coins land heads up given specific conditions for the first two coins: the first coin lands heads up and the second lands tails up.

Identify Possible Outcomes

List the possible outcomes for the three coins under the specified conditions. If the first coin is heads (H) and the second is tails (T), the possible outcomes are: HTH, HTT. We have two outcomes to consider.

Count Favorable Outcomes

We want precisely two coins to be heads up (H). The only possible favorable outcome that meets this condition is HHT.

Find Total Outcomes

Under the given conditions, the possible outcomes are HTH or HTT, making a total of 2 outcomes.

Calculate Probability

To find the probability, we use the formula \( P = \frac{ ext{Number of Favorable Outcomes}}{ ext{Total Possible Outcomes}} \). Here, \( P = \frac{1}{2} \), since there is 1 favorable outcome (HHT) out of 2 possible outcomes.

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with calculating the likelihood of different outcomes. It allows us to predict how likely it is for an event to occur based on the possible scenarios. Probability is usually expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. For example, the probability of flipping a fair coin and getting heads is 0.5 because there are two possible outcomes, heads or tails, each equally likely.

There are several key concepts within probability theory:

• Event: This refers to the outcome or a specific set of outcomes that we are interested in. For instance, getting two heads when tossing three coins is an event.
• Sample Space: This is the set of all possible outcomes of an experiment. In a three-coin toss, this would include all possible sequences like HHH, HHT, HTH, etc.
• Favorable Outcomes: These are the outcomes which match the specific event we are interested in. They help us determine the probability of the event.
• Probability Formula: The probability of an event occurring is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Understanding these concepts is crucial for solving problems related to probability.

Coin Toss Probability

A coin toss is one of the simplest models of probability and provides a perfect example to illustrate basic probability concepts. When we toss a fair coin, we assume it has two equally likely outcomes: heads (H) or tails (T).

In the context of a series of coin tosses, like flipping three coins, we can apply specific conditions to predict probabilities. For example, if the first coin lands heads up and the second lands tails up, we limit our sample space to accommodate these conditions. Thus, we only consider outcomes like HTH and HTT to find out if we can meet additional conditions, such as getting exactly two heads.

• In each coin toss, only two options occur: either heads or tails.
• Multiple coin tosses can be compounded to create a set of probable sequences.
• Coin toss probability can be expanded to include conditions and find probabilities based on restricted outcomes.

This real-world application of probability using coins helps in easily visualizing and calculating probabilities in more complex scenarios.

Counting Outcomes in Probability

Counting outcomes in probability involves understanding and listing all possible occurrences in a random event to precisely calculate probabilities. This exercise includes counting favorable outcomes to determine the likelihood of a specific event happening under given conditions.

For the scenario where three coins are tossed, we might initially have 8 possible outcomes (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT), but conditions can narrow this sample space. When the first two coin tosses are defined as heads and tails respectively (H, T), our outcomes are limited to scenarios fulfilling this requirement, such as HTH and HTT. Therefore, our sample space under these conditions becomes much smaller.

• Sample Space Reduction: With defined conditions, count only the outcomes that fit those specific criteria.
• Favorable Outcome Identification: Identify the specific outcomes that fulfill the complete event requirements (like exactly two heads).
• Application of Counting Principles: Counting techniques, like identifying sequential factors that impact outcomes, aid in simplifying the probability calculation.

Through the careful analysis of all possible outcomes, we can efficiently calculate probabilities, ensuring accurate predictions.