Question

events a and b are independent. the probability of a occurring is \\(\frac{2}{5}\\). the probability of b occurring is \\(\frac{1}{4}\\). what is \\(p(a\\) and b)? \\(\bigcirc\\ \frac{1}{10}\\) \\(\bigcirc\\ \frac{1}{3}\\) \\(\bigcirc\\ \frac{5}{8}\\) \\(\bigcirc\\ \frac{13}{20}\\)

Explanation:

Step1: Recall the formula for independent events

For two independent events \( A \) and \( B \), the probability of both \( A \) and \( B \) occurring is given by the formula \( P(A \cap B)=P(A)\times P(B) \).

Step2: Substitute the given probabilities

We know that \( P(A)=\frac{2}{5} \) and \( P(B)=\frac{1}{4} \). Substituting these values into the formula, we get:
\( P(A \cap B)=\frac{2}{5}\times\frac{1}{4} \)

Step3: Simplify the product

To multiply the fractions, we multiply the numerators and the denominators:
\( \frac{2\times1}{5\times4}=\frac{2}{20}=\frac{1}{10} \)

Answer:

\( \frac{1}{10} \) (corresponding to the first option)