Question

question 4 (1 point)
what is the correct formula for find the probability that two events (a and b) will both happen?
o a ( p ( a \text { and } b ) = p ( a ) + p ( b ) )
o b ( p ( a \text { and } b ) = p ( a ) cdot p ( b ) )
o c ( p ( a \text { and } b ) = \frac { p ( a ) } { p ( b ) } )
o d ( p ( a \text { and } b ) = p ( a ) + p ( b ) - p ( a \text { or } b ) )

Explanation:

Step1: Analyze option a

The formula \( P(A + B)=P(A)+P(B) \) is for the probability of either \( A \) or \( B \) (mutually - exclusive events), not both.

Step2: Analyze option b

For independent events \( A \) and \( B \), the probability that both \( A \) and \( B \) occur is given by the multiplication rule \( P(A\cap B)=P(A)\times P(B) \).

Step3: Analyze option c

The formula \( P(A|B)=\frac{P(A\cap B)}{P(B)} \) (conditional probability) is not the formula for \( P(A\cap B) \) as presented in this option.

Step4: Analyze option d

The formula \( P(A\cup B)=P(A)+P(B)-P(A\cap B) \) is for the probability of \( A \) or \( B \) (inclusion - exclusion principle), not for \( P(A\cap B) \).

Answer:

B. \( P(A \text{ and } B)=P(A)\cdot P(B) \)