Let $F(z)$ be the cumulative density function of the standard normal variate $z$, then which of the following are correct? (A) $F(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^z e^{-\frac{z^2}{2}} dz$, $-\infty < z < \infty$ (B) $F(-z) = 1 - F(z)$ (C) $F(0) = 0$ (D) $F(\infty) = 1$ Choose the correct answer from the options given below:

A man speaks the truth 8 out of 10 times. A die is tossed. He reports that it was 5. What is the probability that it was actually 5?

Probabilities of solving a specific mathematical problem independently by three students are $\frac{1}{2},\frac{1}{3}$ and $\frac{1}{4}$. If all the three students try to solve the problem independently, the probability that the problem is solved, is:

Bag I contains 4 red and 3 black balls and bag II contains 6 red and 5 black balls. One ball is drawn at random from one of the bags, the probability that it is a red ball, is

The probability of not getting $53$ Tuesdays in a leap year is :

Given that the events A and B are mutually exclusive such that $P(A)=\dfrac{1}{2}$, $P(A\cup B)=\dfrac{2}{3}$ and $P(B)=p$, then the value of $p$ is:

If a computer code is correctly programmed, it gives 90% acceptable results. But if it is not correctly programmed, it gives only 40% acceptable results. From previous experience, it is observed that only 80% of codes are correctly programmed. If after a certain programming, the code gives 2 acceptable results, then the approximate probability that the code is correctly programmed is