Problem 95 - 1,001 GED Practice Questions For Dummies

Question

What is the probability of tossing 5 heads in a row when using a fair coin? (A) \(\frac{1}{2}\) (B) \(\frac{1}{8}\) (C) \(\frac{1}{32}\) (D) \(\frac{5}{2}\)

Step-by-Step Solution

Answer

(C) \(\frac{1}{32}\)

Step 1: Understand the concept of a fair coin and independent events

A fair coin has two sides, head and tail, and each side has an equal chance of appearing when the coin is tossed. This means the probability of obtaining a head or a tail is equally likely (\(\frac{1}{2}\) chance). Tossing a fair coin multiple times results in independent events, meaning the outcome of one toss does not affect the outcome of any other toss.

Step 2: Calculate the probability of one head

Since the coin is fair, the probability of getting a head in one toss is \(\frac{1}{2}\).

Step 3: Calculate the probability of 5 heads in a row

In order to calculate the probability of obtaining 5 heads in a row, we must multiply the probabilities of getting a head in each toss. Since these events are independent and each has a probability of \(\frac{1}{2}\), the overall probability can be calculated as follows: Probability of 5 heads in a row = \(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{32}\)

Step 4: Identify the correct answer option

Based on our calculations, the correct probability is \(\frac{1}{32}\). Comparing this with the answer options, we can see that the correct answer is (C) \(\frac{1}{32}\).

Key Concepts

Independent Events

When it comes to probability, understanding the concept of independent events is crucial. Independent events refer to scenarios where the outcome of one event has no impact on the outcome of another. For instance, when tossing a coin, the result of the first toss does not influence the next toss. Each flip remains unaffected by previous results.

This independence is a key reason why we can multiply probabilities to determine the likelihood of multiple events occurring in sequence. When you toss a coin multiple times, each toss is an independent event. This concept is vital in calculating the probabilities of sequences, such as achieving heads in consecutive coin tosses. Remember, in probability, independence means each event starts with a clean slate, uninfluenced by past outcomes.

Fair Coin

A fair coin is a basic yet essential concept in probability. A fair coin has two sides: heads and tails. In the context of probability, each side has an equal chance of landing face up when flipped. This means the probability of getting a head is \( \frac{1}{2} \) and the probability of getting a tail is also \( \frac{1}{2} \).

Understanding what a fair coin means helps simplify problems like determining the probability of multiple coin tosses. Since every toss of a fair coin has the same chance of resulting in heads or tails, it provides a straightforward foundation for calculating more complex probabilities in sequences.

Probability of Multiple Events

Calculating the probability of multiple events, especially when they are independent, involves multiplying the chances of each individual event. Take the example of tossing a fair coin five times and aiming for heads each time. Since each toss is independent, we multiply the probability of getting heads for each toss.

• First toss: Probability of heads = \( \frac{1}{2} \)
• Second toss: Probability of heads = \( \frac{1}{2} \)
• Continue this for the desired number of tosses

The calculation for five consecutive heads looks like this: \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \), which simplifies to \( \frac{1}{32} \).

This technique of multiplying probabilities is powerful in statistics, enabling us to predict outcomes over sequences of events, all while relying on the principle of event independence.