Problem 59 - A First Course in Probability

Question

Independent flips of a coin that lands on heads with probability \(p\) are made. What is the probability that the first four outcomes are (a) \(H, H, H, H ?\) (b) \(T, H, H, H ?\) (c) What is the probability that the pattern \(T, H\) \(H, H\) occurs before the pattern \(H, H, H, H ?\) Hint for part \((c):\) How can the pattern \(H, H, H, H\) occur first?

Step-by-Step Solution

Answer

The probabilities for the given outcomes are as follows: (a) The probability of the first four flips being H, H, H, H is \(p^4\). (b) The probability of the first four flips being T, H, H, H is \((1-p)p^3\). (c) The probability that the pattern T, H, H, H occurs before the pattern H, H, H, H is \((1-p)p^3 + p^4(1-p)\).

Step 1: Part (a): Probability of H, H, H, H outcomes

Since the flips are independent, we can multiply the probabilities of each outcome to find the probability of the entire sequence. The probability that the first four flips are H, H, H, H is given by: \(p \cdot p \cdot p \cdot p = p^4\).

Step 2: Part (b): Probability of T, H, H, H outcomes

Similarly, we can find the probability of this specific sequence by multiplying the probabilities of each outcome: \((1-p) \cdot p \cdot p \cdot p = (1-p)p^3\).

Step 3: Part (c): Probability that the pattern T, H, H, H occurs before H, H, H, H

There are two possibilities for the pattern T, H, H, H to occur before H, H, H, H: Case 1: The first four flips are T, H, H, H. Probability: \((1-p)p^3\) Case 2: The first five flips are H, T, H, H, H. The first flip must be H (with probability \(p\)), and then the next four flips must be T, H, H, H. Probability: \(p (1-p)p^3 = p^4(1-p)\) To find the probability that one of these cases happens, we can add their probabilities because they are two disjoint events: \(P(\text{pattern } T, H, H, H \text{ occurs before pattern } H, H, H, H) = (1-p)p^3 + p^4(1-p)\)

Key Concepts

Independent Events

In probability theory, whether something happens by chance is regulated by the principle of independent events. When you flip a coin, the outcome of each flip does not influence the others. This means each flip is an independent event. Whether you get heads or tails on the first flip doesn’t make the second flip any more or less likely to be heads or tails.

• A simple way to think of this is to imagine resetting probabilities after each event.
• The probability of flipping heads remains constant, regardless of past flips.

Understanding independent events is key because it lets us calculate the probability of sequences by multiplying the individual probabilities.

Sequence Probability

Sequence probability is all about determining how likely a series of events will occur in a specific order. Let's start with the simplest example: flipping a coin. Each flip has a probability, such as landing on heads with probability \(p\). To find the probability of a sequence like getting four heads in a row, you multiply the probabilities of each individual flip happening. For example, if you want four heads in a row, you calculate \(p \cdot p \cdot p \cdot p = p^4\). This multiplication shows how we build up more complex probabilities from simple events. If any one of the coin flips had a different outcome, it alters the entire sequence's probability.

• The act of multiplying probabilities manages related events, building up from independent events.
• Every sequence requires each part to succeed, making calculations straightforward once you grasp the basics of independent events.

Coin Flipping

Coin flipping is a classic example of probability theory in action, frequently used to explain random events. In these exercises, we consider a coin where the probability \(p\) of heads may be different from tails, which is often \(1-p\). This probability may not always be 0.5 (a fair coin), allowing us to explore more advanced scenarios.

• Each coin flip is an independent event, as explained before, making calculations easier by assuming no impact from earlier outcomes.
• The probability of any specific sequence of outcomes, such as \(T, H, H, H\), depends on both \(p\) and \(1-p\).

Coin flipping is therefore a simple yet powerful tool for teaching about probability sequences and patterns.

Pattern Occurrence

In the context of these exercises, pattern occurrence examines how likely a particular sequence of events will arise before another sequence. For instance, we look at whether \(T, H, H, H\) occurs before \(H, H, H, H\). To evaluate this, we consider multiple initial sequences of flips, each one contributing to the total probability through its unique path.

• Each possible path that results in the desired pattern is considered, with its probability calculated independently.
• The solution involves summing the probabilities of distinct paths that meet the criteria, acknowledging these events are mutually exclusive.

By analyzing possible patterns, we grow to understand more complex probability scenarios, piecing together each path’s potential to occur within broader sequences.

Probability of Events

The probability of events tells us how likely something is to happen, influenced by its characteristics and conditions. In flipping coins, this can mean the chance of a sequence occurring first amidst potential other sequences. Here, the outcome relies on understanding both sequence probability and the independence of coin flips.

• Events are often analyzed in isolation and combined into more complex analyses, appreciating the uniqueness of each sequence.
• To conclude an overall probability, we merge separate event probabilities, always checking if they are disjoint (mutually exclusive).

Calculating probabilities allows us to understand random yet structured behavior, giving insight into likely outcomes based on known odds, like \(p\) and \(1-p\).