Select The Correct Answer.A Fair Coin Is Tossed 3 Times In A Row. What Is The Probability That Heads Appears Only On The Last Toss?A. 1 8 \frac{1}{8} 8 1 ​ B. 3 8 \frac{3}{8} 8 3 ​ C. 3 16 \frac{3}{16} 16 3 ​ D. 3 4 \frac{3}{4} 4 3 ​

Introduction

Probability is a fundamental concept in mathematics that deals with the likelihood of an event occurring. In this article, we will explore the concept of probability in the context of coin tosses. Specifically, we will examine the probability of getting heads on the last toss when a fair coin is tossed 3 times in a row.

The Basics of Coin Tosses

When a fair coin is tossed, there are two possible outcomes: heads or tails. The probability of getting heads is 1/2, and the probability of getting tails is also 1/2. Since the coin is fair, the probability of getting heads or tails on any given toss is equal.

The Problem at Hand

We are asked to find the probability that heads appears only on the last toss when a fair coin is tossed 3 times in a row. To solve this problem, we need to consider the possible outcomes of the first two tosses and then determine the probability of getting heads on the last toss.

Analyzing the Possible Outcomes

Let's analyze the possible outcomes of the first two tosses. Since each toss has two possible outcomes (heads or tails), there are a total of 2 x 2 = 4 possible outcomes for the first two tosses.

• Outcome 1: Heads on the first toss and tails on the second toss (H-T)
• Outcome 2: Tails on the first toss and heads on the second toss (T-H)
• Outcome 3: Heads on the first toss and heads on the second toss (H-H)
• Outcome 4: Tails on the first toss and tails on the second toss (T-T)

Determining the Probability of Getting Heads on the Last Toss

Now that we have analyzed the possible outcomes of the first two tosses, we need to determine the probability of getting heads on the last toss. Since the coin is fair, the probability of getting heads on any given toss is 1/2.

For each of the 4 possible outcomes of the first two tosses, there is only one outcome that results in heads on the last toss:

• Outcome 1: H-T (Heads on the last toss)
• Outcome 2: T-H (Heads on the last toss)
• Outcome 3: H-H (Heads on the last toss)
• Outcome 4: T-T (No heads on the last toss)

Since there are 3 outcomes that result in heads on the last toss, the probability of getting heads on the last toss is 3/4.

Conclusion

In conclusion, the probability of getting heads on the last toss when a fair coin is tossed 3 times in a row is 3/4. This result can be obtained by analyzing the possible outcomes of the first two tosses and determining the probability of getting heads on the last toss.

Final Answer

The final answer is $\boxed{\frac{3}{4}}$.

Additional Tips and Tricks

• Understanding the concept of probability: Probability is a fundamental concept in mathematics that deals with the likelihood of an event occurring.
• Analyzing possible outcomes: When analyzing possible outcomes, it's essential to consider all possible scenarios and determine the probability of each outcome.
• Determining the probability of an event: The probability of an event can be determined by analyzing the possible outcomes and determining the number of outcomes that result in the event occurring.

Common Mistakes to Avoid

• Not considering all possible outcomes: When analyzing possible outcomes, it's essential to consider all possible scenarios and determine the probability of each outcome.
• Not determining the probability of an event: The probability of an event can be determined by analyzing the possible outcomes and determining the number of outcomes that result in the event occurring.

Real-World Applications

• Probability in finance: Probability is used extensively in finance to determine the likelihood of investment returns and to manage risk.
• Probability in medicine: Probability is used in medicine to determine the likelihood of disease diagnosis and to develop treatment plans.
• Probability in engineering: Probability is used in engineering to determine the likelihood of system failure and to develop safety protocols.

Conclusion

Q: What is the probability of getting heads on the last toss when a fair coin is tossed 3 times in a row?

A: The probability of getting heads on the last toss when a fair coin is tossed 3 times in a row is 3/4.

Q: How do you calculate the probability of an event?

A: To calculate the probability of an event, you need to analyze the possible outcomes and determine the number of outcomes that result in the event occurring. You can then divide the number of favorable outcomes by the total number of possible outcomes to determine the probability of the event.

Q: What is the difference between a fair coin and an unfair coin?

A: A fair coin is a coin that has an equal probability of landing on heads or tails, while an unfair coin is a coin that has a higher probability of landing on one side than the other.

Q: How do you determine the probability of an event when there are multiple possible outcomes?

A: To determine the probability of an event when there are multiple possible outcomes, you need to analyze each possible outcome and determine the probability of the event occurring in each case. You can then add up the probabilities of each outcome to determine the overall probability of the event.

Q: What is the concept of independent events?

A: Independent events are events that do not affect each other. For example, the outcome of a coin toss is an independent event, as it does not affect the outcome of a subsequent coin toss.

Q: How do you calculate the probability of independent events?

A: To calculate the probability of independent events, you can multiply the probabilities of each event together. For example, if the probability of getting heads on a coin toss is 1/2, and the probability of getting tails on a subsequent coin toss is also 1/2, the probability of getting heads and then tails is (1/2) x (1/2) = 1/4.

Q: What is the concept of conditional probability?

A: Conditional probability is the probability of an event occurring given that another event has occurred. For example, the probability of getting heads on a coin toss given that the previous toss was heads is different from the probability of getting heads on a coin toss given that the previous toss was tails.

Q: How do you calculate the probability of conditional events?

A: To calculate the probability of conditional events, you need to analyze the possible outcomes and determine the probability of the event occurring given that the other event has occurred. You can then use the formula P(A|B) = P(A and B) / P(B) to determine the probability of the event.

Q: What is the concept of Bayes' theorem?

A: Bayes' theorem is a mathematical formula that allows you to update the probability of an event based on new information. It is commonly used in Bayesian statistics to update the probability of a hypothesis based on new data.

Q: How do you apply Bayes' theorem in real-world scenarios?

A: Bayes' theorem can be applied in a variety of real-world scenarios, such as medical diagnosis, quality control, and risk assessment. It allows you to update the probability of a hypothesis based on new data and make more informed decisions.

Conclusion

In conclusion, probability is a fundamental concept in mathematics that deals with the likelihood of an event occurring. By understanding the concepts of probability, independent events, conditional probability, and Bayes' theorem, you can make more informed decisions in a variety of fields, including finance, medicine, and engineering.