In a manufacturing plant, two independent automated assembly units, U_1 and U_2, undergo s… — JAMB UTME

In a manufacturing plant, two independent automated assembly units, $U_1$ and $U_2$ , undergo safety inspection. The probability that unit $U_1$ fails the inspection is $\frac{1}{5}$ , while the probability that unit $U_2$ fails the inspection is $\frac{1}{4}$ . What is the probability that at least one of the two units fails the inspection? Express your answer as a decimal.

Answer: 0.4

Answer

0.4

The probability that at least one unit fails is determined using the addition law for non-mutually exclusive independent events:

P(U_1 \cup U_2) = P(U_1) + P(U_2) - P(U_1 \cap U_2) = 0.20 + 0.25 - (0.20 \times 0.25) = 0.45 - 0.05 = 0.40

. Alternatively, using the complement rule gives

1 - P(\text{neither fails}) = 1 - (1 - 0.20)(1 - 0.25) = 1 - 0.60 = 0.40

Step-by-Step Solution

Identify the individual failure probabilities

P(U_1) = 0.20

and

P(U_2) = 0.25

These values represent the single event failure probabilities given in the problem statement.

Calculate the probability that neither unit fails

P(U_1') \times P(U_2') = (1 - 0.20) \times (1 - 0.25) = 0.80 \times 0.75 = 0.60

Because the units operate independently, their complement events (passing inspection) are also independent.

Determine the probability of at least one unit failing

1 - 0.60 = 0.40

The event that at least one unit fails is the complementary event of neither unit failing.

Key Concept

Addition and Multiplication Laws of Probability for Independent Compound Events

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic