A balanced coin is tossed 10 times. Let X denote the number of heads. Calculate the following:
E(X)
To calculate the expected value E(X), we use the formula:
E(X) = Σ(x * P(X = x))
where x is the value of the random variable (in this case, the number of heads) and P(X = x) is the probability of getting x heads.
Since the coin is balanced, the probability of getting a head or a tail is 0.5. Therefore, the probability of getting x heads in 10 tosses follows a binomial distribution with n = 10 and p = 0.5.
Using the binomial probability formula:
P(X = x) = C(n, x) * p^x * (1-p)^(n-x)
= C(10, x) * 0.5^x * 0.5^(10-x)
= C(10, x) * 0.5^10
where C(n, x) is the binomial coefficient.
Now, let's calculate E(X):
E(X) = Σ(x * P(X = x))
= Σ(x * C(10, x) * 0.5^10)
= 0 * C(10, 0) * 0.5^10 + 1 * C(10, 1) * 0.5^10 + 2 * C(10, 2) * 0.5^10 + ... + 10 * C(10, 10) * 0.5^10
After calculating the individual terms and summing them up, we get:
E(X) = 5
Therefore, the expected value of the number of heads in 10 tosses of a balanced coin is 5.