In a telecommunications network, two independent relay switches, R_1 and R_2, operate duri… — JAMB UTME

In a telecommunications network, two independent relay switches, $R_1$ and $R_2$ , operate during a data transmission. The probability that switch $R_1$ functions successfully is $0.80$ , and the probability that switch $R_2$ functions successfully is $0.75$ . What is the probability that at least one of the two switches functions successfully during the transmission?

Cevap: 0.95

Cevap

The probability that at least one switch functions successfully is 0.95.

Since the two switches operate independently, the probability of both failing is the product of their individual failure probabilities:

(1 - 0.80) \times (1 - 0.75) = 0.20 \times 0.25 = 0.05

. Therefore, the probability that at least one switch functions successfully is

1 - 0.05 = 0.95

. Alternatively, using the addition law for independent events:

P(R_1 \cup R_2) = P(R_1) + P(R_2) - P(R_1 \cap R_2) = 0.80 + 0.75 - (0.80 \times 0.75) = 1.55 - 0.60 = 0.95

Adım Adım Çözüm

Determine the probabilities of individual switch failure.

P(R_1') = 0.20 and P(R_2') = 0.25

The probability of an event failing is the complement of its success probability: P(E') = 1 - P(E).

Compute the probability that both switches fail simultaneously.

P(R_1' ∩ R_2') = 0.20 × 0.25 = 0.05

Since the switches operate independently, the multiplication law for independent events applies: P(A ∩ B) = P(A) × P(B).

Calculate the probability that at least one switch functions successfully.

P(at least one) = 1 - 0.05 = 0.95

The complement of 'neither switch functioning' is 'at least one switch functioning'.

Anahtar Kavram

Probability laws for independent compound events and the complement rule

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