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

The probability that a randomly chosen factor of $10^{19}$ is a multiple of $10^{15}$ is

A man is known to speak the truth on an average 4 out of 5 times. He throws a die and reports that it is a five. The probability that it is actually a five 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 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

In a given village there are only three sizes of families: families with 2 members, families with 4 members and families with 6 members. The proportion of families with 2, 4 and 6 members are roughly equal. A poll is conducted in this village wherein a person is chosen at random and asked about his/her family size. The average family size computed by sampling 1000 such persons from the village would be closest to