### Multiplication law of probability

Question Sample Titled 'Multiplication law of probability'

 ${\left({a}\right)}$ Independent events If ${A}$ and ${B}$ are two independent events, then ${P}{\left({A}\cap{B}\right)}={P}{\left({A}\right)}\times{P}{\left({B}\right)}$ . In general, If ${A}_{{1}}$ , ${A}_{{2}}$ , ${A}_{{3}}$ , … , ${A}_{{n}}$ are independent events, then ${P}{\left({A}_{{1}}\cap{A}_{{2}}\cap{A}_{{3}}\cap\right.}$ … $\cap$ ${A}_{{n}}{)}={P}{\left({A}_{{1}}\right)}\times{P}{\left({A}_{{2}}\right)}\times{P}{\left({A}_{{3}}\right)}\times$ … $\times{P}{\left({A}_{{n}}\right)}$ .  ${\left({b}\right)}$ Dependent events If A and B are two dependent events, then  ${P}{\left({A}\cap{B}\right)}$ $={P}{\left({A}\right)}\times{P}{\left({B}{\mid}{A}\right)}$ ...${\left(\ast\right)}$ From ${\left(\ast\right)}$ , when ${P}{\left({A}\right)}\ne{0}$ , the conditional probability ${P}{\left({B}{\mid}{A}\right)}$ can be found by ${P}{\left({B}{\mid}{A}\right)}$ $=\dfrac{{{P}{\left({A}\cap{B}\right)}}}{{{P}{\left({A}\right)}}}$ Note: ${P}{\left({B}{\mid}{A}\right)}$ denotes 'the probability that event B occurs given that event A has occurred'.

 Example A box contains ${3}$ red balls (${R}$) and ${5}$ white balls (${W}$). Find the probabilities of getting two red balls if the balls are drawn one by one at random

 (a) with replacement, (b) without replacement.

 Solution (a) ${P}{\left({R}{R}\right)}=\dfrac{{3}}{{8}}\times\dfrac{{3}}{{8}}=\dfrac{{9}}{{64}}$ (b) ${P}{\left({R}{R}\right)}=\dfrac{{3}}{{8}}\times\dfrac{{2}}{{7}}=\dfrac{{3}}{{28}}$

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