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Probability — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSProbability3 Marks
Question
A box contains 30 bolts and 40 nuts. Half of the bolts and half of the nuts are rusted. If two items are drawn at random, what is the probability that either both are rusted or both are bolts?
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Answer

Box 30-bolts 40-nuts Half the bolts and nuts are rusted $\therefore$ rusted bolts = 15 $\therefore$ rusted bolts = 20 Since two items are drawn $\therefore\text{n}(\text{S})=^{70}\text{C}_2$ Let A be the event of choosing rusting item $\therefore\text{p}(\text{A})=\frac{^{35}\text{C}_2}{^{70}\text{C}_2}$ $=\frac{35\times34}{70\times69}$ Let B be the event of choosing bolts $\therefore\text{p}(\text{B})=\frac{^{30}\text{C}_2}{^{70}\text{C}_2}$ $=\frac{30\times29}{70\times69}$ Also, $\text{n}(\text{A}\cap\text{B})=15$ [bolts that are rusted] $\therefore\text{p}(\text{A}\cap\text{B})=\frac{^{15}\text{C}_2}{^{70}\text{C}_2}{}$ $=\frac{15\times14}{70\times69}$ $\therefore\text{p}(\text{A}\cup\text{B})=\text{p}(\text{A})+\text{p}(\text{B})-\text{p}(\text{A}\cap\text{B})$ $=\frac{35\times34}{70\times69}+\frac{30\times29}{70\times69}-\frac{15\times14}{70\times69}$ $=\frac{1850}{4830}$ $=\frac{185}{483}$

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