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1. It is always true because there is a 50% chance that it will rain tomorrow, and there is also a 50% chance that it will rain after tomorrow.
2. It is sometimes true because if there are a greater number of pupils in a school than the days in a year, then it will always be true. The smaller the number of pupils in a school, the smaller the chance that two pupils will share the same birthday.
3. It is sometimes true. If there are more than 365 pupils in a school, there is a small chance that it will be true because there is a chance that two or more pupils will have the same birthday.
4. It will be sometimes true. This is because the randomly selected person from London could be 100 and the randomly selected person from Calcutta could be 1.
5. It will sometimes be true because the whole class could throw a double six and all be winners, or only two people could throw a double six and both be winners. There is a very small chance that there will be only one winner.
6. It will sometimes be true because there will be an equal chance that both will be thrown, but it will not be necessarily so.
7. It will sometimes be true, but not always because even though there is an equal 50% chance that both will be thrown, it will not always be true.
8. It is sometimes true because the set of numbers 1, 2, 3, 4, 5 have a mode of 3, a median of 3 and modes of 1, 2, 3, 4 and 5. Therefore they have the same averages of 3.
9. The statement will not be true because the mean will always be less than the mode.
10. It is not always true because if six students took the test and two got 1 and the others got 7, the average mark is 30/6 which is equal to five. Therefore it is only sometimes true that half the people taking a test will score below the average mark.
11. It is sometimes true because if all of the students achieved the same mark, they did not score above the average, but the same as it.
12. It is not always true because if you scored 220 points over a series of 20 games, then the average score would be 11. It is sometimes true because if you got 320 points over a series of 16 games you would get an average of 20.