Probability Part 3

Concept:Use mean and variance formulas to find unknown observations. Then find probability using complementary event.Explanation:Given mean = 8, so $\frac{2+4+10+x+12+14+y}{7} = 8$.Simplify: $2+4+10+12+14 = 42$, so $42 + x + y = 56$.Thus $x + y = 14$.Given variance = 16, formula: $\frac{\sum x_i^2}{7} - (\text{mean})^2 = 16$.Compute squares: $2^2=4$, $4^2=16$, $10^2=100$, $12^2=144$, $14^2=196$. Sum = $4+16+100+144+196 = 460$.So $460 + x^2 + y^2 = 7(16+64) = 7 \times 80 = 560$.Therefore $x^2 + y^2 = 100$.Now $x+y=14$ and $x^2+y^2=100$. Solve: $(x+y)^2 = x^2+y^2+2xy$ gives $196 = 100+2xy$, so $2xy=96$, $xy=48$.Thus $x$ and $y$ are roots of $t^2 -14t +48=0$: $(t-6)(t-8)=0$, so $(x,y)=(8,6)$ since $x>y$.New set: $\{1,2,3, x-4, y, 5\} = \{1,2,3,4,6,5\}$ which is $\{1,2,3,4,5,6\}$.We want probability that the smaller of two numbers chosen without replacement is less than 4.Probability = $1 - P(\text{both numbers} \geq 4)$.Numbers $\geq 4$ in set: $\{4,5,6\}$ (3 numbers). Total numbers: 6.$P(\text{both} \geq 4) = \frac{{}^3C_2}{{}^6C_2} = \frac{3}{15} = \frac{1}{5}$.Thus required probability = $1 - \frac{1}{5} = \frac{4}{5}$.Answer:$\frac{4}{5}$ (Option A)