Two independent weather forecasting stations, A and B, operate in a region. The probabilit… — JAMB UTME

Two independent weather forecasting stations, $A$ and $B$ , operate in a region. The probability that station $A$ makes an accurate forecast on any given day is $0.80$ , and the probability that station $B$ makes an accurate forecast is $0.75$ . What is the probability that at least one of the two stations makes an accurate forecast on a given day?

Cevap: 0.95

Cevap

The probability that at least one of the two stations makes an accurate forecast is

0.95

The probability of at least one event occurring is given by

P(A \cup B) = P(A) + P(B) - P(A \cap B)

. Because events

A

and

B

are independent,

P(A \cap B) = P(A) \times P(B) = 0.80 \times 0.75 = 0.60

. Substituting into the addition rule yields

0.80 + 0.75 - 0.60 = 0.95

. Alternatively, using the complement rule:

1 - P(A')P(B') = 1 - (1 - 0.80)(1 - 0.75) = 1 - (0.20 \times 0.25) = 1 - 0.05 = 0.95

Adım Adım Çözüm

Calculate the probability of both events occurring simultaneously using the multiplication law for independent events.

P(A \cap B) = P(A) \times P(B) = 0.80 \times 0.75 = 0.60

Since the two forecasting stations operate independently, the joint probability is the product of their individual probabilities.

Apply the general addition law of probability to calculate the probability of at least one station making an accurate forecast.

P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.80 + 0.75 - 0.60 = 0.95

The probability of compound event 'at least one' corresponds to the union of the two events.

Anahtar Kavram

Addition and Multiplication Laws of Probability for Independent Events

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