Ten eggs are drawn successively, with replacement, from a basket containing $10\%$ defective eggs. The probability that there is at least one defective egg is $1-{\left(\frac{9}{10}\right)}^k$, then find the value of $k$.

Probability of getting $5$ exactly twice in $7$ throws of a die is $\frac{a}{12}\times {\left(\frac{5}{6}\right)}^5$. Find $a$.

A bag contains $5$ white, $7$ red and $8$ black balls. If $4$ balls are drawn one by one with replacement, then what is the probability that none is white.

A bag contains $5$ white, $7$ red and $8$ black balls. If $4$ balls are drawn one by one with replacement, then what is the probability that all are white.

A bag contains $5$ white, $7$ red and $8$ black balls. If $4$ balls are drawn one by one with replacement, then what is the probability that only $2$ are whites?

A box contains $100$ tickets, each bearing a different number from $1$ to $100$. If 5 tickets are drawn successively with replacement from the box, then the probability that all tickets bear numbers divisible by $10$ is ${\left(\frac{1}{10}\right)}^k$, then find the value of $k$.

If the chance that any of the five telephone lines is busy at an instant is $0.01$, then what is the probability that all lines are busy? What is the probability that more than $3$ lines are busy?

The probability of a man hitting a target is $0.25$. He shoots $7$ times. What is the probability of his hitting at least twice?

During a war,$1$ in $10$ ships was sunk in the average in making a certain voyage. What is the probability that at least $3$ out of a convoy of $6$ ships would arrive safely?