Consider
1- $\lambda_1<0<\lambda_2<\ldots<\lambda_n$ are some real scalars.
2- $x_i$'s are real variables satisfying $\sum_{i=1}^nx_i^2=1$
what can we say about the probability of points $(x_1,x_2,\ldots,x_n)$ in $\mathbb{R}^n$ satisfying the following equation:
$\sum_{i=1}^n\lambda_i^2x_i^2-(\sum_{i=1}^n\lambda_ix_i^2)^2\geq-\frac{\lambda_1\lambda_n}{2}$
can we say that the probability of points satisfying this inequality increases by growing $n$? why? can we compute or approximate this probability?