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\text{ and }b)\\)?\
\\(\frac{1}{10}\\)\
\\(\frac{1}{3}\\)\
\\(\frac{5}{8}\\)\
\\(\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 \( P(A \text{ and } B)=P(A)\times P(B) \).

Step2: Substitute the given values into the formula

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

$$ P(A \text{ and } B)=\frac{2}{5}\times\frac{1}{4} $$

Step3: Calculate the product

To multiply the fractions, we multiply the numerators together and the denominators together:

$$ \frac{2\times1}{5\times4}=\frac{2}{20}=\frac{1}{10} $$

Answer:

\(\frac{1}{10}\) (Note: Assuming the first option is \(\frac{1}{10}\) (maybe a typo in the original where it's written as \(\frac{1}{10}\) instead of \(\frac{1}{10}\) with correct formatting), so the correct option is the first one: \(\frac{1}{10}\))