Consider the following events:
$E_1$: Six fair dice are rolled and at least one die shows six.
$E_2$: Twelve fair dice are rolled and at least two dice show six.
Let $p_1$ be the probability of $E_1$ and $p_2$ be the probability of $E_2$. Which of the following is true?

Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting a head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$,respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice,independently,and $\beta$ is the number of heads that appear when $C_2$ is tossed twice,independently. Then,the probability that the roots of the quadratic polynomial $x^2 - \alpha x + \beta$ are real and equal is:

In a box containing $100$ apples,$10$ are defective. The probability that in a sample of $6$ apples,$3$ are defective is

For a binomial variate $X$, if $n=4$ and $P(X=4)=6 P(X=2)$, then the value of $p$ is

$A$ box contains $20\%$ defective bulbs. Five bulbs are chosen randomly from this box. The probability that exactly $3$ of the chosen bulbs are defective is

It is observed that $25\%$ of the cases related to child labour reported to the police station are solved. If $6$ new cases are reported,then the probability that at least $5$ of them will be solved is . . . . . .