The probability that a blue-eyed person is left-handed is 1/7. The probability that a left-handed person is blue-eyed is 1/3. The probability that a person has neither of these attributes is 4/5. What is the probability that a person has both attributes?

Question

The probability that a blue-eyed person is left-handed is  1/7. The probability that a left-handed person is blue-eyed is  1/3. The probability that a person has neither of these attributes is  4/5. What is the probability that a person has both attributes?

turingtest · Answer

the probability that someone has at least one attribute is$$P(at~least~one~attribute)=1-P(neither~attribute)$$let $A=left-handed$ and $B=right-handed$
We can then write this as
$$P(A\cup B)=1-P(A^c\cap B^c)$$you can then apply the inclusion-exclusion principle to find the probability of the intersection $P(A\cap B)$ (the probability of being both right-handed and left-handed):$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$