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JEE Main 20248 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
8 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Probability

JEE Main 2024 · 8 April · Morning Shift · Mathematics · Probability

Three balls are drawn at random from a bag containing 5 blue and 4...

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables XX and YY respectively denote the number of blue and yellow balls. If Xˉ\bar{X} and Yˉ\bar{Y} are the means of XX and YY respectively, then 7Xˉ+4Yˉ7 \bar{X}+4 \bar{Y} is equal to________Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables X and Y

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Step-by-step solution

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables X and Y
7x‾=5C14C2+5C2⋅4C1×2+5C3⋅4C0×39C3×730+80+3084×7=14012=706=353\begin{aligned} & 7 \overline{\mathrm{x}}=\frac{{ }^5 \mathrm{C}_1{ }^4 \mathrm{C}_2+{ }^5 \mathrm{C}_2 \cdot{ }^4 \mathrm{C}_1 \times 2+{ }^5 \mathrm{C}_3 \cdot{ }^4 \mathrm{C}_0 \times 3}{{ }^9 \mathrm{C}_3} \times 7 \\ & \frac{30+80+30}{84} \times 7 \\ & =\frac{140}{12}=\frac{70}{6}=\frac{35}{3}\end{aligned}
\begin{array}{|l|l|l|l|l|l|}<br />\hline yellow & 0 & 1 & 2 & 3 & 4 \\<br />\hline & & { }^5 \mathrm{C}_2{ }^4 \mathrm{C}_1 & { }^5 \mathrm{C}_1{ }^4 \mathrm{C}_2 & { }^5 \mathrm{C}_0{ }^4 \mathrm{C}_3 & 0 \\<br />\hline<br />\end{array}
4yˉ=40+60+1284×4=11221=1634 \bar{y}=\frac{40+60+12}{84} \times 4=\frac{112}{21}=\frac{16}{3}
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