Probability & Strategy Puzzles

A coin has been flipped 5 times in a row, landing heads every time. Sameer thinks it is "due" for tails now, so the next flip is more likely to be tails than heads. His sister says both are still equally likely. Who is right?

In a lottery you pick 6 numbers. Dev says picking 1-2-3-4-5-6 is foolish, because such a neat set never wins; a scattered set like 4-11-17-28-33-41 is more likely. His friend says both sets have exactly the same chance. Who is right?

Two dice are rolled together. Ved thinks getting a sum of 2 and getting a sum of 7 are equally likely, since 2 through 12 are just "one number each". His teacher says getting 7 is far more likely than getting 2. Who is right?

The forecast says a 50% chance of rain on Saturday and a 50% chance on Sunday. Tanya says that means it is 100% certain to rain on at least one day of the weekend. Her brother says the real chance is lower than that. Who is right?

A game show has 3 doors — one hides a prize, two are empty. You pick a door. The host (who knows where the prize is) opens one of the other two, revealing it is empty, and asks: switch, or stay? Ayush thinks the two remaining doors now each have an equal 50-50 chance, so switching gives no advantage. His teacher says switching is actually better. Who is right?

Of two doors, one leads to freedom, the other to danger. A guard stands at each door — one always tells the truth, the other always lies, but you do not know which is which. You may ask only one guard a single question. Meet thinks that without knowing who the liar is, there is no way to be certain of the right door. His sister says there is one clever question that always works. Who is right?