Two candidates, Ada and Babatunde, independently sit for an aptitude test. The probability… — JAMB UTME

Two candidates, Ada and Babatunde, independently sit for an aptitude test. The probability that Ada passes the test is $\frac{1}{3}$ and the probability that Babatunde passes is $\frac{2}{5}$ . What is the probability that at least one of them passes the test?

A
$\frac{2}{15}$
B
$\frac{2}{5}$
$\frac{3}{5}$ Cevap
D
$\frac{11}{15}$

Cevap

The probability that at least one candidate passes the test is

\frac{3}{5}

For independent events, the probability of both occurring is given by

P(A \cap B) = P(A) \cdot P(B) = \frac{1}{3} \cdot \frac{2}{5} = \frac{2}{15}

. Applying the addition law

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

yields

\frac{1}{3} + \frac{2}{5} - \frac{2}{15} = \frac{9}{15} = \frac{3}{5}

. Alternatively, calculating

1 - P(\text{neither passes}) = 1 - (1 - \frac{1}{3})(1 - \frac{2}{5}) = 1 - (\frac{2}{3} \cdot \frac{3}{5}) = 1 - \frac{2}{5} = \frac{3}{5}

gives the identical correct outcome.

Adım Adım Çözüm

Calculate the probability that both Ada and Babatunde pass the test.

P(\text{Ada} \cap \text{Babatunde}) = \frac{1}{3} \times \frac{2}{5} = \frac{2}{15}

Since the two events are independent, the multiplication law of probability applies:

P(A \cap B) = P(A) \times P(B)

Apply the general addition law of probability to find the union of the two events.

P(\text{Ada} \cup \text{Babatunde}) = P(\text{Ada}) + P(\text{Babatunde}) - P(\text{Ada} \cap \text{Babatunde})

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

Substitute the probabilities and solve for the outcome.

P(\text{Ada} \cup \text{Babatunde}) = \frac{1}{3} + \frac{2}{5} - \frac{2}{15} = \frac{5 + 6 - 2}{15} = \frac{9}{15} = \frac{3}{5}

Find a common denominator (15) and simplify the resulting fraction to its lowest terms.

Anahtar Kavram

Probability Laws for Compound and Independent Events

Tahmini Süre:1m 30s

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