A farmer plants two types of crops, maize and cassava, in separate fields. The probability… — JAMB UTME

A farmer plants two types of crops, maize and cassava, in separate fields. The probability that the maize crop yields a successful harvest is $\frac{3}{8}$ , and the probability that the cassava crop yields a successful harvest is $\frac{1}{2}$ . Assuming that the harvest outcomes of the two crops are independent, what is the probability that at least one of the crops yields a successful harvest?

A
$\frac{3}{16}$
$\frac{11}{16}$ Cevap
C
$\frac{7}{8}$
D
$\frac{5}{16}$

Cevap

The probability that at least one of the crops yields a successful harvest is

\frac{11}{16}

The probability of at least one successful harvest is given by the union of the two events,

P(M \cup C) = P(M) + P(C) - P(M \cap C)

. Since the harvests are independent, the joint probability is

P(M \cap C) = P(M) \times P(C) = \frac{3}{8} \times \frac{1}{2} = \frac{3}{16}

. Substituting these values gives

\frac{3}{8} + \frac{1}{2} - \frac{3}{16} = \frac{6}{16} + \frac{8}{16} - \frac{3}{16} = \frac{11}{16}

Adım Adım Çözüm

Identify given probabilities and event relationship

Let

M

be maize success and

C

be cassava success:

P(M) = \frac{3}{8}

and

P(C) = \frac{1}{2}

. The events are independent.

Establishing the basic event probabilities and independence condition is essential for applying compound probability laws.

Calculate the joint probability of both events occurring

P(M \cap C) = P(M) \times P(C) = \frac{3}{8} \times \frac{1}{2} = \frac{3}{16}

For independent events, the multiplication law states that the probability of both occurring is the product of their individual probabilities.

Apply the addition law of probability for the union of two events

P(M \cup C) = P(M) + P(C) - P(M \cap C) = \frac{3}{8} + \frac{1}{2} - \frac{3}{16} = \frac{6}{16} + \frac{8}{16} - \frac{3}{16} = \frac{11}{16}

The probability that at least one crop succeeds corresponds to the union

P(M \cup C)

, which requires subtracting the intersection to prevent double counting.

Anahtar Kavram

Addition Law of Probability and Multiplication Law for Independent Events

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