The binomial distribution is used to calculate the probability that a sample will contain a certain number of a certain type of item. In this case, the binomial distribution is used to calculate th

a. The Binomial distribution can be stated as follows: B(n, p) where n is the number of trials (in this case, the sample size, which is 20) and p is the probability of success on each trial (in this case, the proportion of green beads, which is 0.45 or 45%). So, the Binomial distribution for the random variable is B(20, 0.45). b. The probability that the sample contains fewer than 12 green beads is found by summing the probabilities of getting exactly 0, 1, 2, ..., up to 11 green beads. However, calculating this by hand is tedious and sometimes prone to error. Instead, many people would use a binomial probability table or a binomial calculator to find this probability. You can use a binomial calculator (which can be found online) as follows: Enter n = 20 (number of trials), p = 0.45 (probability of success on each trial), and x = 11 (number of successes), and then calculate the cumulative probability for x less than or equal to 11 to find the probability of getting fewer than 12 green beads. Remember that 'fewer than 12' includes 11, so in a calculator or table you would look for the cumulative probability up to and including 11, not 10. In the interest of providing a complete response, however, be aware that the exact calculation without using a binomal calculator would look something like the summation of [20Cx * (0.45)^x * (0.55)^(20-x)] from x=0 to 11, where 20Cx represents the number of combinations of 20 things taken x at a time. This calculation is often beyond what's expected for a non-math major.