Final answer:
When tossing 3 coins, the probability distribution results in a mean of 1.5, a variance of 0.75, and a standard deviation of approximately 0.866, which quantifies the variability of the outcomes.
Step-by-step explanation:
When tossing 3 coins, the possible outcomes and probability distribution for the number of heads (X) can be represented as follows:
- 0 heads (TTT): P(X=0) = (1/2)^3 = 1/8
- 1 head (HTT, THT, TTH): P(X=1) = 3/8
- 2 heads (HHT, HTH, THH): P(X=2) = 3/8
- 3 heads (HHH): P(X=3) = 1/8
The mean (μ) is calculated as:
μ = Σ[X * P(X)] = (0 * 1/8) + (1 * 3/8) + (2 * 3/8) + (3 * 1/8) = 1.5
The variance (σ^2) can be found using:
σ^2 = Σ[(X - μ)^2 * P(X)] = [(0 - 1.5)^2 * 1/8] + [(1 - 1.5)^2 * 3/8] + [(2 - 1.5)^2 * 3/8] + [(3 - 1.5)^2 * 1/8] = 0.75
The standard deviation (σ) is the square root of the variance:
σ = √σ^2 = √0.75 ≈ 0.866
To summarize, in flipping 3 coins, we have a mean of 1.5 heads, a variance of 0.75, and a standard deviation of approximately 0.866. These calculations help express the quantitative disorder, or variability, of our random variable X, the number of heads.