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Chapter 5: Problem 11

What is the probability of obtaining five heads in a row when flipping a coin? Interpret this probability.

Short Answer

Expert verified
The probability of obtaining five heads in a row is \( \frac{1}{32} \), which is about 0.03125.

Step by step solution

01

- Determine the probability of a single event

A fair coin has two sides: heads (H) and tails (T). The probability of getting heads in a single flip is \( \frac{1}{2} \).
02

- Identify the number of events

We are looking for the probability of obtaining 5 heads in a row. Therefore, we have 5 independent events (coin flips).
03

- Use the multiplication rule for independent events

Since these events are independent, the combined probability of all events occurring is the product of their individual probabilities. Thus, the probability of getting five heads in a row is \[ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{32} \].
04

- Interpret the result

The probability of obtaining five heads in a row when flipping a coin is \( \frac{1}{32} \). This value indicates that such an outcome is quite rare.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
In probability, independent events are those whose outcomes do not affect each other. This means the result of one event has no impact on the result of another.
When flipping a coin, each flip is independent of the others.
The probability of getting heads or tails remains the same for every flip.
For example, getting heads on one flip doesn't affect whether you'll get heads or tails on the next flip.
Multiplication Rule in Probability
The multiplication rule in probability helps us find the probability of multiple independent events occurring together.
To use this rule, multiply the probability of each individual event.
If you are flipping a coin five times and want to get heads each time, you use the multiplication rule.
For each flip, the probability of getting heads is \(\frac{1}{2}\).
Therefore, the combined probability of getting five heads in a row is: \[ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{32} \].
Probability Interpretation
Interpreting probability helps you understand how likely an event is to occur.
When you compute the probability of flipping five heads in a row, you get \(\frac{1}{32}\).
This fraction means that out of 32 tries, you can expect this outcome to happen once.
Another way to see this is that the probability is 3.125%.
Knowing this helps you realize that getting five heads in a row is quite rare.
Coin Flipping Probability
Coin flipping is a simple yet powerful example in probability.
A fair coin has two outcomes: heads or tails, each with a probability of \(\frac{1}{2}\).
When flipping a coin multiple times, each flip is independent, meaning earlier flips don't affect later ones.
For complex events, like getting five heads in a row, using the multiplication rule makes calculating the probability straightforward.
Coin flipping problems are a great way to practice and understand basic probability concepts.

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Most popular questions from this chapter

Suppose a compact disk (CD) you just purchased has 13 tracks. After listening to the CD, you decide that you like 5 of the songs. The random feature on your CD player will play each of the 13 songs once in a random order. Find the probability that among the first 4 songs played (a) you like 2 of them; (b) you like 3 of them; (c) you like all 4 of them.

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Suppose you are dealt 5 cards from a standard 52 -card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering the following questions: (a) How many ways can 5 cards be selected from a 52 card deck? (b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck? (c) The remaining 2 cards must be different from the 3 chosen and different from each other. For example, if we drew three kings, the 4 th card cannot be a king. After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. If we have three kings, then we can choose twos, threes, and so on. Of the 12 ranks remaining, we choose 2 of them and then select one of the 4 cards in each of the two chosen ranks. How many ways can we select the remaining 2 cards? (d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not like?
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