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

Explanation

To determine the number of points in the sample space for the experiment where a coin is tossed 10 times, we can follow these steps:

### Step 1: Understand the Experiment
When a coin is tossed, there are two possible outcomes for each toss: heads (H) or tails (T). In this experiment, we are interested in the difference between the number of heads and the number of tails after 10 tosses.

### Step 2: Define the Variables
Let:
- \( H \) = number of heads
- \( T \) = number of tails

Since the coin is tossed 10 times, we have:
\[ H + T = 10 \]

### Step 3: Express the Outcome
The outcome of the experiment is defined as:
\[ \text{Outcome} = H - T \]

### Step 4: Relate Heads and Tails
From the equation \( H + T = 10 \), we can express \( T \) in terms of \( H \):
\[ T = 10 - H \]

Substituting this into the outcome equation gives:
\[ \text{Outcome} = H - (10 - H) = 2H - 10 \]

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

### Step 6: Calculate the Corresponding Outcomes
Now we can calculate the corresponding outcomes for each value 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 7: List the Possible Outcomes
The possible outcomes from the experiment are:
\[ -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10 \]

### Step 8: Count the Points in the Sample Space
The total number of distinct outcomes is 11 (from -10 to 10, inclusive).

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

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