Question

True/False: You are more likely to get a pattern of HTHHHTHTTH than HHHHHHHHTT when you flip a coin 10 times.

Short answer

False, both sequences have equal probability of occurring.

Step-by-step solution

Understand the Problem

When flipping a coin 10 times, we are looking at the probability of getting specific sequences (HTHHHTHTTH and HHHHHHHHTT), which means counting the ways each sequence can occur.

Recognize Each Flip's Probability

Each flip of a fair coin has two possible outcomes: Heads (H) or Tails (T). Thus, the probability of any specific sequence of outcomes occurring in a series of flips is determined by the individual probabilities of each flip.

Calculate Probability of Any 10-Flip Sequence

The probability of obtaining any specific sequence of 10 coin flips is \[\left(\frac{1}{2}\right)^{10} = \frac{1}{1024}.\]This is because each coin flip is independent, and there are two possible outcomes per flip.

Conclude Equivalence of Sequences

Both sequences HTHHHTHTTH and HHHHHHHHTT are specific sequences of 10 flips. Therefore, each has a probability of \(\frac{1}{1024}\) of occurring. Thus, neither sequence is more likely than the other.

Key concepts

Independent Events

When we refer to independent events in probability, we mean that the outcome of one event does not affect the outcome of another. This is a crucial concept to understand when dealing with coin flips. Each time you flip a coin, the result—either heads (H) or tails (T)—does not influence the next flip. This independence implies that each flip has no memory and cannot predict or affect the future flips.

In the problem of flipping a coin 10 times, each flip is an independent event. Whether the first flip comes out heads or tails has no bearing on whether the second flip will do the same. This characteristic of coin flips ensures that each sequence of flips, such as HTHHHTHTTH or HHHHHHHHTT, is equally probable. Recognizing the independence of each flip simplifies the calculation of probabilities for sequences of outcomes.

Coin Flip

Flipping a coin is one of the most basic examples of a random experiment, commonly used in probability theory. A coin flip typically has two possible outcomes: heads (H) or tails (T). The probability of getting heads in a single coin flip is \(rac{1}{2}\), and similarly, the probability of getting tails is also \(rac{1}{2}\). This makes a fair coin flip a perfect model for understanding randomness and probability.

When you flip a coin multiple times, the outcomes of each flip accumulate to form a sequence. For example, a sequence of 10 flips could yield outcomes like HTHHHTHTTH or HHHHHHHHTT. Understanding the fundamental 50/50 chance for each flip aids in calculating the likelihood of any specific sequence of outcomes over a series of flips. Each specific sequence occurs with a probability of \(rac{1}{1024}\), highlighting the concept of extremely low odds when exact repetitions are demanded.

Specific Sequences

Specific sequences in the context of coin flips refer to predetermined sets of outcomes stemming from multiple flips. Let's say you want an exact sequence of heads and tails, like HTHHHTHTTH. Each individual flip's outcome is independent and follows a simple probability rule, but when looking to achieve a specific sequence, the complexity seems to increase.

Yet, despite this perceived complexity, the real probability associated with any specific sequence in a set of 10 coin flips remains straightforward. Since each flip is independent and comes with a \(rac{1}{2}\) chance, the probability for any sequence of 10 flips becomes \(\left(\frac{1}{2}\right)^{10} = \frac{1}{1024}\). This logic holds true regardless of whether the sequence looks random like HTHHHTHTTH or regular like HHHHHHHHTT. Therefore, every specific combination has the same chance of appearing in a random batch of flips, demonstrating the inherent fairness and unpredictability of chance events.