The probability that a randomly chosen positive divisor of $10 ^ {2023}$ is an integer multiple of $10 ^ {2001}$ is

From a pack of 52 cards, we draw one by one, without replacement. If $f(n)$ is the probability that an Ace will appear at the $n^{\text{th}}$ turn, then

A fair die is rolled repeatedly. The probability that the cumulative sum is at least 17 in the third trial is

A and B take part in a rifle shooting match. The probability of A hitting the target is 0.4, while the probability of B hitting the target is 0.6. If A has the first shot, post which both strike alternately, then the probability that A hits the target before B hits it is

The minimum number of times a fair coin must be tossed so that the probability of getting at least one head exceeds 0.8 is

Three dice are thrown simultaneously and the sum of the three numbers appearing on the top faces of the dice is found to be 10. The probability that these three numbers are distinct, is

A die is thrown three times and the sum of the three numbers is found to be 15. The probability that the first throw was a four is