A un-fair coin is flipped 3 times, the probability of getting a head is 1/5 . find the probability distribution of resulting  head(s) by finishing the table below.

Please show your answer to 4 decimal places
k	P(X = k)
0	
1	
2	
3	


Use the probability distribution above, and what you have learned in the previous section, please find the mean and standard deviation. Please show your answer to 1 decimal place.

 


### Step 1: Probability Distribution Table

Let's denote the number of heads in 3 flips as \( X \). The probability of getting a head on a single flip is \( P(H) = \frac{1}{5} = 0.2 \), and the probability of getting a tail is \( P(T) = 1 - 0.2 = 0.8 \).

We can use the binomial distribution to calculate the probability of getting \( k \) heads in 3 flips:

\[
P(X = k) = \binom{3}{k} \times (0.2)^k \times (0.8)^{3-k}
\]

Where \( \binom{3}{k} \) is the binomial coefficient, calculated as \( \frac{3!}{k!(3-k)!} \).

#### Calculation for each \( k \):

1. **For \( k = 0 \) (0 heads):**
   \[
   P(X = 0) = \binom{3}{0} \times (0.2)^0 \times (0.8)^3 = 1 \times 1 \times 0.512 = 0.5120
   \]

2. **For \( k = 1 \) (1 head):**
   \[
   P(X = 1) = \binom{3}{1} \times (0.2)^1 \times (0.8)^2 = 3 \times 0.2 \times 0.64 = 0.3840
   \]

3. **For \( k = 2 \) (2 heads):**
   \[
   P(X = 2) = \binom{3}{2} \times (0.2)^2 \times (0.8)^1 = 3 \times 0.04 \times 0.8 = 0.0960
   \]

4. **For \( k = 3 \) (3 heads):**
   \[
   P(X = 3) = \binom{3}{3} \times (0.2)^3 \times (0.8)^0 = 1 \times 0.008 \times 1 = 0.0080
   \]

#### Completed Probability Distribution Table:

\[
\begin{array}{|c|c|}
\hline
k & P(X = k) \\
\hline
0 & 0.5120 \\
1 & 0.3840 \\
2 & 0.0960 \\
3 & 0.0080 \\
\hline
\end{array}
\]

### Step 2: Mean (Expected Value) and Standard Deviation

#### Mean (Expected Value) \( \mu \):

\[
\mu = E(X) = \sum_{k=0}^{3} k \times P(X = k)
\]

\[
\mu = 0 \times 0.5120 + 1 \times 0.3840 + 2 \times 0.0960 + 3 \times 0.0080
\]

\[
\mu = 0 + 0.3840 + 0.1920 + 0.0240 = 0.6000
\]

So, the mean (expected value) \( \mu = 0.6 \).

#### Variance \( \sigma^2 \) and Standard Deviation \( \sigma \):

First, calculate the variance \( \sigma^2 \):

\[
\sigma^2 = E(X^2) - (E(X))^2
\]

Where:

\[
E(X^2) = \sum_{k=0}^{3} k^2 \times P(X = k)
\]

\[
E(X^2) = 0^2 \times 0.5120 + 1^2 \times 0.3840 + 2^2 \times 0.0960 + 3^2 \times 0.0080
\]

\[
E(X^2) = 0 + 0.3840 + 0.3840 + 0.0720 = 0.8400
\]

Now, calculate the variance:

\[
\sigma^2 = 0.8400 - (0.6)^2 = 0.8400 - 0.36 = 0.4800
\]

Finally, the standard deviation \( \sigma \):

\[
\sigma = \sqrt{0.4800} \approx 0.6928
\]

Rounding to 1 decimal place:

\[
\sigma \approx 0.7
\]

### Final Answers

- **Mean (Expected Value) \( \mu \):** 0.6
- **Standard Deviation \( \sigma \):** 0.7

Would you like more details or have any questions?

---

Here are some related questions:

1. How would the distribution change if the coin were fair?
2. What is the probability of getting exactly 2 heads in 5 flips with the same coin?
3. How would the expected value change if the probability of heads was \( \frac{2}{5} \)?
4. What is the variance of the distribution if the coin is flipped 4 times instead of 3?
5. How does the standard deviation change with an increasing number of flips?

**Tip:** Understanding the binomial distribution is crucial for calculating probabilities in scenarios involving independent events, like coin flips.

A un-fair coin is flipped 3 times, the probability of getting a head is 1/5. Find the probability distribution of resulting head(s) by finishing the table below:

k    P(X = k)
0    
1    
2    
3    

Use the probability distribution above, and what you have learned in the previous section, please find the mean and standard deviation. Please show your answer to 1 decimal place.

Find the probability distribution of heads in 3 flips of an unfair coin with P(Heads) = 1/5. This problem explores the binomial distribution, mean, and standard deviation, showing detailed steps for calculating probabilities and statistics. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation