Question

event a and b are such as that p(a)=1/2 and p(b)=7/12 and p(not a or not b ) =1/4. state whether a and b are independent​

Answer 1

Answer:

The events A and B are not independent.

Step-by-step explanation:

The information provided is:

$P (A) =\frac{1}{2}\\ P (B) = \frac{7}{12}\\ P(A^{c} \cup B^{c})=\frac{1}{4}$

The probability of event (A ∪ B) is:

$P(A\cup B)=1-P (A^{c}\cup B^{c})=1-\frac{1}{4} =\frac{3}{4}$

The addition rule of probability is:

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

Compute the probability of the even (A ∩ B) as follows:

$P(A\cup B)=P(A)+P(B)-P(A\cap B)\\\frac{3}{4} =\frac{1}{2}+\frac{7}{12} -P(A\cap B)\\P(A\cap B)=\frac{1}{2}+\frac{7}{12}-\frac{3}{4}\\=\frac{1}{3}$

If events A and B are independent then, $P(A\cap B)=P(A)\times P(B)$

Compute the product of the probabilities of event A and B as follows:

$P(A)\times P(B)=\frac{1}{2}\times \frac{7}{12}=\frac{7}{24}$

The value of $P(A\cap B)\neq P(A)\times P(B)$

Thus, the events A and B are not independent.