Mathematical Methods Β· Unit 1 Β· Probability Β· Conditional probability and independence
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A researcher is studying the reliability of two independent safety sensors in a manufacturing plant. Sensor A detects faults with probability 0.92, and Sensor B detects faults with probability 0.88. (a) Calculate the probability that both sensors detect a fault in the same inspection cycle. (1 mark) (b) Calculate the probability that at least one sensor detects a fault. (2 marks)
A quality control analyst is testing two independent production lines, Line A and Line B. The probability that Line A produces a defective component is 0.08, and the probability that Line B produces a defective component is 0.05. (a) Calculate the probability that both lines produce a defective component in the same time period. (2 marks) (b) Calculate the probability that at least one of the two lines produces a defective component in the same time period. (2 marks)
A retailer sells two independent electronic devices: a tablet and a smartwatch. The probability that a randomly selected tablet has a fault is P(T) = 0.08. The probability that a randomly selected smartwatch has a fault is P(S) = 0.06. (a) Use the multiplication rule for independent events to find the probability that both devices have a fault. (1) (b) Find the probability that at least one device does not have a fault. (2)
A basket contains 8 red balls and 12 blue balls. Two balls are drawn at random from the basket with replacement. Determine the probability that both balls drawn are red.