Final answer:
To find the probability P(X≤μₓ), where X is a binomial random variable with n=20 and p=0.5, use the normal approximation to the binomial distribution. Calculate the mean (μ) and standard deviation (σ) of the binomial distribution, then use the normal distribution with mean μ and standard deviation σ to find P(X≤μₓ).
Step-by-step explanation:
To find the probability P(X≤μₓ), where X is a binomial random variable with n=20 and p=0.5, we can use the normal approximation to the binomial distribution. First, calculate the mean (μ) and standard deviation (σ) of the binomial distribution, where μ = np and σ = √(npq). In this case, μ = 20*0.5 = 10 and σ = √(20*0.5*0.5) = 2.5. To find P(X≤μₓ), we use the normal distribution with mean μ and standard deviation σ.
P(X≤μₓ) = P(X≤10) ≈ P(Z≤(10-0.5-0.5)/2.5), where Z is a standard normal random variable.
Using a standard normal table or a calculator, we can find P(Z≤(10-0.5-0.5)/2.5) ≈ P(Z≤(9/2.5)) ≈ P(Z≤3.6) ≈ 0.999.