SOLUTION: Consider the experiment of throwing two coins. a. establish the probability distribution b. determine the expected value c. determine the standard deviation

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Question 1184134: Consider the experiment of throwing two coins.
a. establish the probability distribution
b. determine the expected value
c. determine the standard deviation
Answer by CPhill(2264) (Show Source):
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Here's how to analyze the experiment of throwing two coins:
**a. Probability Distribution:**
First, let's list the sample space (all possible outcomes):
* HH (Heads, Heads)
* HT (Heads, Tails)
* TH (Tails, Heads)
* TT (Tails, Tails)
Since the coins are fair, each outcome has an equal probability of 1/4.
Let X be the random variable representing the number of heads. The possible values of X are 0, 1, and 2. We can now create the probability distribution:
* P(X = 0) = P(TT) = 1/4
* P(X = 1) = P(HT) + P(TH) = 1/4 + 1/4 = 2/4 = 1/2
* P(X = 2) = P(HH) = 1/4
Here's the probability distribution in table form:
| X (Number of Heads) | P(X) |
|---|---|
| 0 | 1/4 |
| 1 | 1/2 |
| 2 | 1/4 |
**b. Expected Value (E[X] or μ):**
The expected value is calculated as:
E[X] = Σ [x * P(X = x)]
E[X] = (0 * 1/4) + (1 * 1/2) + (2 * 1/4)
E[X] = 0 + 1/2 + 1/2
E[X] = 1
The expected number of heads is 1.
**c. Standard Deviation (σ):**
1. **Calculate the variance (Var[X] or σ²):**
Var[X] = Σ [(x - E[X])² * P(X = x)]
Var[X] = (0 - 1)² * 1/4 + (1 - 1)² * 1/2 + (2 - 1)² * 1/4
Var[X] = 1 * 1/4 + 0 * 1/2 + 1 * 1/4
Var[X] = 1/4 + 0 + 1/4
Var[X] = 1/2 = 0.5
2. **Calculate the standard deviation (σ):**
σ = √Var[X]
σ = √(1/2)
σ ≈ 0.707
The standard deviation of the number of heads is approximately 0.707.