How to Solve Discrete Random Variables — Univariate RV — binomial distribution Questions on Exam P
Sample Practice Problem
ID: #WRBRN
Let \(X_{1}\) and \(X_{2}\) be the completion counts for two independent calibration teams that each test 10 sensors. Each sensor completes calibration with probability 70%. Calculate the probability that exactly one team has at least 9 completions.
📖 Worked Solution & Strategy
For each team, completion count \(X_{i}\) follows \(\operatorname{Binomial}(10, \frac{7}{10})\). The probability of at least 9 completions is \(h = \Pr(X_{i} \ge 9) = \binom{10}{9}(\frac{7}{10})^9(1-\frac{7}{10})^1 + \binom{10}{10}(\frac{7}{10})^{10}(1-\frac{7}{10})^0 = 0.1493\). Because the two teams are independent, team outcomes multiply. Exactly one team succeeds if Team 1 succeeds while Team 2 fails, or vice versa: \(\Pr(\text{exactly 1}) = h(1-h) + (1-h)h = 2h(1-h) = 2(0.1493)(1-0.1493) = 0.2540\).