A coin is tossed 10 times. The number of heads minus the number of tails in 10 tosses is considered as the outcome of the experiment. What is the number of points in the sample space?

Options

A. 10
B. 11
C. 21
D. 99

Answer

To solve the problem of finding the number of points in the sample space when a coin is tossed 10 times, we will follow these steps:

### Step 1: Understand the Experiment
When a coin is tossed 10 times, the possible outcomes can be represented by the number of heads (H) and the number of tails (T). Since each toss can result in either heads or tails, we can calculate the difference between the number of heads and tails.

### Step 2: Define the Outcome
The outcome of the experiment is defined as the number of heads minus the number of tails. Mathematically, this can be expressed as:
\[ \text{Outcome} = H - T \]
Since \( H + T = 10 \) (the total number of tosses), we can rewrite T as \( T = 10 - H \). Therefore, the outcome becomes:
\[ \text{Outcome} = H - (10 - H) = 2H - 10 \]

### Step 3: Determine the Range of Heads
The number of heads (H) can range from 0 to 10, inclusive. Therefore, we can list the possible values of H:
- If \( H = 0 \): Outcome = \( 2(0) - 10 = -10 \)
- If \( H = 1 \): Outcome = \( 2(1) - 10 = -8 \)
- If \( H = 2 \): Outcome = \( 2(2) - 10 = -6 \)
- If \( H = 3 \): Outcome = \( 2(3) - 10 = -4 \)
- If \( H = 4 \): Outcome = \( 2(4) - 10 = -2 \)
- If \( H = 5 \): Outcome = \( 2(5) - 10 = 0 \)
- If \( H = 6 \): Outcome = \( 2(6) - 10 = 2 \)
- If \( H = 7 \): Outcome = \( 2(7) - 10 = 4 \)
- If \( H = 8 \): Outcome = \( 2(8) - 10 = 6 \)
- If \( H = 9 \): Outcome = \( 2(9) - 10 = 8 \)
- If \( H = 10 \): Outcome = \( 2(10) - 10 = 10 \)

### Step 4: List All Possible Outcomes
From the calculations above, the possible outcomes are:
- -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10

### Step 5: Count the Unique Outcomes
Now, we count the unique outcomes:
- The outcomes are: -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10
- This gives us a total of 11 unique outcomes.

### Conclusion
Thus, the number of points in the sample space is **11**.