Question

What is the probability of getting tails 6 consecutive times when you toss a coin (assume the coin has no bias)? A. 1/4 B. 1/8 C. 1/16 D. 1/32 E. 1/64

Short answer

E. 1/64

Step-by-step solution

Understand the Problem

This exercise asks to find the probability of flipping a coin and getting tails 6 times in a row. Each flip is an independent event with a fixed probability.

Probability of a Single Event

The probability of getting tails on a single coin flip is 1/2. This is because there are only two possible outcomes: heads or tails.

Calculate Probability for Multiple Events

To find the probability of getting tails 6 consecutive times, multiply the probability of getting tails on each flip together: \[\text{Probability} = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \]

Simplify the Calculation

Simplify the multiplication: \[\text{Probability} = \frac{1}{2^6} = \frac{1}{64} \]

Choose the Correct Option

The correct option among the choices provided is E. The probability of getting tails 6 consecutive times is 1/64.

Key concepts

Independent Events

In probability, an event refers to the outcome of a random process. The term 'independent events' means that the outcome of one event does not affect the outcome of another event.
This principle is fundamental when dealing with consecutive coin flips.
Each flip of a coin has no influence on the results of any subsequent flips.
If a coin shows tails once, this should not affect the chance of landing tails again on the next flip. Every coin flip stands alone regarding probability.

Fixed Probability

When we say a coin has no bias, each side (heads or tails) is equally likely to land face up.
This means each outcome has a fixed probability. For a fair coin, the probability of getting tails on a single flip is always \( \frac{1}{2} \), or 50%.
No matter how many times you flip the coin, this probability remains unchanged.
Fixed probability helps in predicting the likelihood of outcomes in multiple trials.

Multiplying Probabilities

To find the probability of multiple independent events happening consecutively, you multiply the individual probabilities of each event.
For example, the probability of getting tails 6 times in a row involves multiplying the probability of getting tails in each single flip:
\[ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{64} \]
Multiplication simplifies the process of understanding multiple consecutive outcomes.

Simplifying Fractions

Simplifying fractions is a crucial step in deriving clean and understandable results. The initial multiplication in the example gave us \[ \frac{1}{2^6} \]
Instead of handling repeated multiplication, it's easier to simplify directly:
Two raised to the power of six is 64, hence \[ \frac{1}{64} \]
This makes it easy to recognize which multiple-choice option is correct and quickly understand the probability.