Final answer:
To find the exact value of P(x ≥ 11) for a binomial distribution with n = 20 and p = 0.4, use a statistical calculator function such as binomcdf to calculate the complementary probability P(x ≤ 12) and subtract this from 1.
Step-by-step explanation:
The binomial probability distribution is a tool for calculating the likelihood of a fixed number of successes in a certain number of trials, with each trial having two potential outcomes (success or failure) and the same probability of success.
In this case, the given values are n = 20 (number of trials) and p = 0.4 (probability of success on any individual trial). To calculate the probability P(x ≥ 11), we are interested in the occurrence of 11 or more successes out of 20 trials.
Since calculating this directly can be tedious, we rely on the cumulative distribution function provided by a statistical calculator or software. Using the TI-83/84 calculator, for example, we would employ the binomcdf function to determine P(x ≤ 12), as this is the probability of 12 or fewer successes, which is complementary to our desired P(x ≥ 11).
After calculating P(x ≤ 12) using the calculator instruction binomcdf(20, 0.4, 12), we would then subtract this value from 1 to find the exact value of P(x ≥ 11).