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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

According to the U.S. Bureau of Labor Statistics, there is a \(5.84 \%\) probability that a randomly selected employed individual has more than one job (a multiple-job holder). Also, there is a \(52.6 \%\) probability that a randomly selected employed individual is male, given that he has more than one job. What is the probability that a randomly selected employed individual is a multiple-job holder and male? Would it be unusual to randomly select an employed individual who is a multiple-job holder and male?

Exclude leap years from the following calculations and assume each birthday is equally likely: (a) Determine the probability that a randomly selected person has a birthday on the 1 st day of a month. Interpret this probability. (b) Determine the probability that a randomly selected person has a birthday on the 31 st day of a month. Interpret this probability. (c) Determine the probability that a randomly selected person was born in December. Interpret this probability. (d) Determine the probability that a randomly selected person has a birthday on November \(8 .\) Interpret this probability. (e) If you just met somebody and she asked you to guess her birthday, are you likely to be correct? (f) Do you think it is appropriate to use the methods of classical probability to compute the probability that a person is born in December?

Suppose a single card is selected from a standard 52-card deck. What is the probability that the card drawn is a king? Now suppose a single card is drawn from a standard 52-card deck, but we are told that the card is a heart. What is the probability that the card drawn is a king? Did the knowledge that the card is a heart change the probability that the card was a king? What is the term used to describe this result?

In \(2004,17 \%\) of Florida's population was 65 and over (Source: U.S. Census Bureau). What is the probability that a randomly selected Floridian is 65 or older?

In the Big Game, an urn contains balls numbered 1 to \(50,\) and a second urn contains balls numbered 1 to \(36 .\) From the first urn, 5 balls are chosen randomly, without replacement. From the second urn, 1 ball is chosen randomly. For a \(\$ 1\) bet, a player chooses one set of five numbers to match the balls selected from the first urn and one number to match the ball selected from the second urn. To win, all six numbers must match; that is, the player must match the first 5 balls selected from the first urn and the single ball selected from the second urn. What is the probability of winning the Big Game with a single ticket?
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