At a carnival, there's a unique coin that is three times more likely to land on heads compared to tails. If this coin is tossed twice, what is the probability that exactly one of the tosses results in heads?

To solve this problem, let's define the probabilities and then calculate the required probability.

### Step 1: Define the probabilities
- Let \( P(H) \) be the probability that the coin lands on heads.
- Let \( P(T) \) be the probability that the coin lands on tails.

Given that the coin is three times more likely to land on heads than on tails, we can write:
\[
P(H) = 3P(T)
\]
Since the total probability must equal 1 (i.e., \( P(H) + P(T) = 1 \)), we can substitute \( P(H) = 3P(T) \) into this equation:
\[
3P(T) + P(T) = 1
\]
\[
4P(T) = 1
\]
\[
P(T) = \frac{1}{4}
\]
\[
P(H) = 3P(T) = 3 \times \frac{1}{4} = \frac{3}{4}
\]

So, the probability of heads is \( \frac{3}{4} \) and the probability of tails is \( \frac{1}{4} \).

### Step 2: Calculate the probability of exactly one heads in two tosses

There are two possible outcomes where exactly one of the tosses results in heads:
1. The first toss is heads and the second toss is tails.
2. The first toss is tails and the second toss is heads.

We calculate the probability for each case:
- **Case 1: Heads on the first toss, tails on the second toss.**
  \[
  P(H \text{ first and } T \text{ second}) = P(H) \times P(T) = \frac{3}{4} \times \frac{1}{4} = \frac{3}{16}
  \]
- **Case 2: Tails on the first toss, heads on the second toss.**
  \[
  P(T \text{ first and } H \text{ second}) = P(T) \times P(H) = \frac{1}{4} \times \frac{3}{4} = \frac{3}{16}
  \]

### Step 3: Add the probabilities of the two cases

Since these two cases are mutually exclusive, the total probability of exactly one heads in two tosses is:
\[
P(\text{Exactly one heads}) = \frac{3}{16} + \frac{3}{16} = \frac{6}{16} = \frac{3}{8}
\]

### Final Answer
The probability that exactly one of the two tosses results in heads is:

\[
\boxed{\frac{3}{8}}
\]

Would you like more details or have any questions? Here are some related questions:

1. What is the probability that both tosses result in heads?
2. What is the probability that both tosses result in tails?
3. How would the probability change if the coin was twice as likely to land on heads compared to tails?
4. What is the probability that at least one toss results in heads?
5. If the coin is tossed three times, what is the probability of getting exactly two heads?
6. How would you calculate the expected number of heads in 10 tosses of this coin?
7. What is the probability of getting no heads in two tosses?
8. Can you generalize the probability for any biased coin being tossed \( n \) times?

**Tip:** When dealing with probability problems involving multiple outcomes, consider each possible scenario, calculate its probability, and then combine the results as appropriate.

Learn how to calculate the probability of exactly one heads in two tosses of a coin biased to land on heads three times more frequently than tails. This problem involves applying probability theory and conditional probability concepts. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation