A coin tossed five times. What is the probability that heads are observed more than three times?

Options

A. `(3)/(16)`
B. `(5)/(16)`
C. `(1)/(2)`
D. `(3)/(32)`

Answer

To find the probability of observing heads more than three times when a coin is tossed five times, we can use the binomial probability formula. Here’s a step-by-step solution:

### Step 1: Define the problem
We need to find the probability of getting more than 3 heads when a coin is tossed 5 times. This means we need to calculate the probabilities for getting 4 heads and 5 heads.

### Step 2: Identify parameters
In this scenario:
- Number of trials (n) = 5 (since the coin is tossed 5 times)
- Probability of getting heads (p) = 1/2
- Probability of getting tails (q) = 1/2

### Step 3: Use the binomial probability formula
The binomial probability formula is given by:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]
where:
- \( \binom{n}{k} \) is the binomial coefficient (n choose k)
- \( k \) is the number of successful outcomes (in this case, heads)

### Step 4: Calculate for k = 4
First, calculate the probability of getting exactly 4 heads:
\[ P(X = 4) = \binom{5}{4} \left(\frac{1}{2}\right)^4 \left(\frac{1}{2}\right)^{5-4} \]
\[ = \binom{5}{4} \left(\frac{1}{2}\right)^4 \left(\frac{1}{2}\right)^1 \]
\[ = 5 \cdot \left(\frac{1}{2}\right)^5 \]
\[ = 5 \cdot \frac{1}{32} = \frac{5}{32} \]

### Step 5: Calculate for k = 5
Next, calculate the probability of getting exactly 5 heads:
\[ P(X = 5) = \binom{5}{5} \left(\frac{1}{2}\right)^5 \left(\frac{1}{2}\right)^{5-5} \]
\[ = 1 \cdot \left(\frac{1}{2}\right)^5 \]
\[ = 1 \cdot \frac{1}{32} = \frac{1}{32} \]

### Step 6: Add the probabilities
Now, we sum the probabilities of getting 4 heads and 5 heads:
\[ P(X > 3) = P(X = 4) + P(X = 5) \]
\[ = \frac{5}{32} + \frac{1}{32} = \frac{6}{32} \]
\[ = \frac{3}{16} \]

### Step 7: Conclusion
Thus, the probability of observing heads more than three times when a coin is tossed five times is:
\[ \boxed{\frac{3}{16}} \]