Let "X" be the number of defective microscopes. Because the events (finding that certain microscope is defecrive or not) have equal probability "p = 1.7 \\cdot {10^{ - 3}}" that it will be defecrive, we can conclude that "X" follows the binomial distribution "X \\sim B(n,p)" (where "n = 50") with probability mass function
where "C_n^k = \\frac{{n!}}{{k!(n - k)!}}" is the binomial coefficient.
Under "one of the first" they mean "exactly one", we need to find the probability "P(X = 1)", thus
"P(X = 1) = {p_X}(1) = \\frac{{50!}}{{1!49!}} \\cdot 1.7 \\cdot {10^{ - 3}} \\cdot {(1 - 1.7 \\cdot {10^{ - 3}})^{49}} \\approx 0.0782008"If they mean "one or more", we need to find the probability "P(X \\geqslant 1)" that can be rewritten as
and
thus
Answer: "P(X = 1) \\approx 0.0782008".
Comments
Leave a comment