https://file.aiquickdraw.com/mathbot/file/1729053459266h3lu4591.jpg

Let's break down the problem in the image step by step:

### Problem:
You are asked to toss two fair coins (say a dime and a quarter), with the possible outcomes for each coin being either heads (H) or tails (T). Based on this, you are asked to provide:

1. **The sample space** (a)
2. **The probability of getting heads on the dime** (b)
3. **The probability of getting heads on the quarter** (c)
4. **The probability of getting both heads** (d)
5. **The probability of getting the same outcome on both coins** (e)

### Solutions:

**(a) The Sample Space:**
The sample space consists of all possible outcomes of the two coin tosses. Since each coin can result in either heads (H) or tails (T), the possible combinations for two coins are:
- \( (H, H) \): Heads on both
- \( (H, T) \): Heads on the first coin (dime), tails on the second (quarter)
- \( (T, H) \): Tails on the first coin (dime), heads on the second (quarter)
- \( (T, T) \): Tails on both coins

So, the sample space \( S \) is:
\[ S = \{ (H, H), (H, T), (T, H), (T, T) \} \]

**(b) The Probability of Heads on the Dime:**
The dime can be either heads or tails, and since it’s a fair coin, the probability of getting heads on the dime is:
\[ P(H \text{ on dime}) = \frac{1}{2} \]

**(c) The Probability of Heads on the Quarter:**
Similarly, the quarter can also be either heads or tails. Since it's a fair coin, the probability of getting heads on the quarter is:
\[ P(H \text{ on quarter}) = \frac{1}{2} \]

**(d) The Probability of Getting Both Heads:**
Getting both heads (i.e., \( (H, H) \)) is one specific outcome from the sample space. Since there are 4 equally likely outcomes in total (as shown in the sample space), the probability of getting both heads is:
\[ P(\text{Both heads}) = \frac{1}{4} \]

**(e) The Probability of Getting the Same Outcome on Both Coins:**
The "same outcome" means either both heads (\( H, H \)) or both tails (\( T, T \)). There are two favorable outcomes: \( (H, H) \) and \( (T, T) \). Out of the 4 possible outcomes in the sample space, the probability of getting the same outcome on both coins is:
\[ P(\text{Same outcome}) = \frac{2}{4} = \frac{1}{2} \]

---

Would you like more detailed explanations on any of these parts or further clarifications?

### Expanding questions:
1. What if the coins were not fair? How would the probabilities change?
2. How can we calculate the probability of getting at least one tail?
3. What is the complement of the event of getting both heads?
4. How would the sample space change if three coins were tossed instead of two?
5. What is the expected value of the number of heads when two coins are tossed?

### Tip:
In probability problems, identifying the sample space correctly is the key to finding accurate probabilities for different events.

Two fair coins are tossed (a dime and a quarter). The two possible outcomes for a single coin are heads (h) and tails (t). Give each of the following:
(a) the sample space
(b) the probability of heads on the dime
(c) the probability of heads on the quarter
(d) the probability of getting both heads
(e) the probability of getting the same outcome on both coins

Explore the probability of various outcomes when two fair coins, a dime and a quarter, are tossed. Understand sample space, the probability of heads, and the chance of getting both heads or the same outcome. Suitable for middle 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