Final answer:
The probability of a binomially distributed random variable X being greater than 10, when n=20 and p=0.5, can be deduced to be close to 0.5 due to the symmetry of the distribution.
Step-by-step explanation:
If X follows a binomial distribution B(20, 0.5), to find the probability of X being greater than 10, we first understand that the binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, with the same probability of success on each trial. To find P(X > 10), we can do one of two things: calculate P(X = 11) + P(X = 12) + ... + P(X = 20) directly using the binomial probability formula or use the complement rule where P(X > 10) = 1 - P(X ≤ 10).
Typically, calculating the probability for each individual outcome and adding them up can be time-consuming. If the binomial distribution parameters meet certain criteria, we can approximate this with a normal distribution, but in this case, since our probability of success p is 0.5, the distribution is symmetric, and we can simply deduce that P(X > 10) would be the same as P(X < 10) because of this symmetry and knowing that the sum of probabilities is 1. Therefore, P(X > 10) would be close to 0.5, and in this case, since we only have discrete choices, the closest option is option d) 0.500