Question

In three flips of a coin, what is the probability that two flips in a row are heads, given that there is an even number of heads?

Short answer

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

Step-by-step solution

List Possible Outcomes

First, determine all possible outcomes for flipping a coin three times. Each flip can result in either heads (H) or tails (T), so the possibilities are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. There are a total of 2^3 = 8 possible outcomes.

Filter Outcomes with Even Number of Heads

Next, identify the scenarios with an even number of heads. The count of heads in each outcome should be 0 or 2. These outcomes are: HTT (1 head, not valid), THT (1 head, not valid), TTH (1 head, not valid), TTT (0 heads, valid), HHT (2 heads, valid), HTH (2 heads, valid), THH (2 heads, valid), and HHH (3 heads, not valid). So the valid outcomes are: TTT, HHT, HTH, THH.

Find Outcomes with Two Consecutive Heads

From the valid outcomes,THT, HHT, HTH, and THH, identify those with two consecutive heads. These outcomes are: HHT and THH.

Calculate Probabilities

Determine the probability of having two consecutive heads, given there is an even number of heads. We found 4 outcomes with an even number of heads (TTT, HHT, HTH, THH), and 2 of these have two consecutive heads (HHT and THH). Thus, the probability is the ratio of these 2 outcomes to the 4 possible outcomes, which is \( \frac{2}{4} = \frac{1}{2} \).

Key concepts

Coin Flipping

Coin flipping is a simple yet powerful illustration of probability. Each flip of a coin results in one of two outcomes: heads (H) or tails (T). The outcomes of a series of coin flips can be depicted in a sequence, like HHH or TTH.

When you flip a coin three times, there are several possible sequences you can get. With each flip being independent, the total number of outcomes is determined by the formula \(2^n\), where \(n\) is the number of flips. For three flips, this results in \(2^3 = 8\) outcomes.

Here, we consider all combinations of coin flipping as equally likely. This assumption is crucial for calculating probabilities in exercises like finding outcomes that meet specific criteria.

Conditional Probability

Conditional probability is the likelihood of an event occurring, given that another event has already occurred. It's denoted by \(P(A \mid B)\), read as "the probability of \(A\) given \(B\)."

In our exercise, we need to find the probability of getting two consecutive heads, given that there is an even number of heads. This simplifies our sample space to only those outcomes that already satisfy the 'even number of heads' condition.

Using conditional probability, we assess only the relevant outcomes and calculate the ratio between outcomes where two consecutive heads appear and the total of even-headed outcomes. The probability tells us the likelihood within this constrained scenario.

Combinatorics

Combinatorics involves counting, arranging, and structuring items in specific ways. It's a key concept in probability calculations.

In this exercise, we used combinatorics when listing all the possible outcomes of coin flips and then filtered them based on specified criteria, like having two consecutive heads and an even number of heads. This method ensures that we consider only the relevant scenarios for our probability calculation.

Combinatorics helps us break down complex problems into simpler parts, making it easier to find patterns, calculate probabilities, and solve the problems systematically. Understanding it enriches your ability to tackle a wide range of probability-based questions.