# If A and B are two events associated with a random experiment

Question:

If A and B are two events associated with a random experiment such that P(A) = 0.25, P(B) = 0.4 and P(A or B) = 0.5, find the values of

(i) $P(A$ and $B)$

(ii) $\mathrm{P}(\mathrm{A}$ and $\overline{\mathrm{B}})$

Solution:

(i) Given : P(A) = 0.25, P(A or B) = 0.5 and P(B) = 0.4

To find : P(A and B)

Formula used : P(A or B) = P(A) + P(B) - P(A and B)

Substituting in the above formula we get,

$0.5=0.25+0.4-P(A$ and $B)$

$0.5=0.65-P(A$ and $B)$

$P(A$ and $B)=0.65-0.5$

$P(A$ and $B)=0.15$

$\mathrm{P}(\mathrm{A}$ and $\mathrm{B})=0.15$

(ii) Given : $P(A)=0.25, P(A$ and $B)=0.15($ from part (i))

To find : $\mathrm{P}(\mathrm{A}$ and $\bar{B})$

Formula used : $\mathrm{P}(\mathrm{A}$ and $\bar{B})=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A}$ and $\mathrm{B})$

Substituting in the above formula we get

$P(A$ and $\bar{B})=0.25-0.15$

$\mathrm{P}(\mathrm{A}$ and $\bar{B})=0.10$

$\mathrm{P}(\mathrm{A}$ and $\bar{B})=0.10$