Answer: 1) The game is not fair, 2) The game is fair
Explanation:
1) In the case of the coin, we will assume that it is a fair coin and there is an equal probability that the coin will land either heads or tails on each flip. In other words, we are assuming that flipping the coin is a completely random process.
When flipping a coin twice, there are 4 distinct possibilities, where H means the coin lands head side up and T means it lands tails side up. The first letter represents the first flip of the coin and the second represents the second flip. The possible outcomes are:
HH, HT, TH, TT
As you can see, there is only 1 possible outcome where the coin will land tails up twice, while there are 2 possible outcomes where the coin will land heads up once and tails up once. Since the game doesn't specify the order in which the coin needs to land on either side for player 2 to win, both of these possibilities are fair game.
Now, according to our earlier assumption that the process of flipping the coin is truly random, we can say that there is a 50/50, chance of the coin landing on either side before we flip it. There is also a 100% chance that, in 2 flips, the coin behaves according to one of the possibilities stated above. This means that each possibility has an equal 25% chance of happening. We determined there are 2 possible outcomes where the coin lands on both a head and a tail, opposed to the 1 possible outcome where the coin lands tails up on both flips. Adding the probabilities, this means that the second player has a 50% chance of winning the game while the first player only has a 25% chance of winning the game, and thus - it is not fair.
2) In the second game, we will also assume that the process of rolling the dice is completely random and there is an equal chance that it lands on each side. The possible outcomes for both dice combining to sum to 5 are:
1,4; 4,1; 2,3 and 3;2
and the possible outcomes for both dice combining to sum to 9 are:
4,5; 5,4; 2,6 and 6,2
Assuming the dice are normal, 6-sided dice, there is a 1/6 or a 16.67% chance of the dice landing on each side. The outcomes on these rolls are dependent on one another and represent compound probabilities. in other words, the probability of rolling a 2 on a single dice is 16.67%, and the probability of rolling a 6 on a single dice is also 16.67%, but the probability of rolling a 2 on one dice and (key word here being and) a 6 on the other dice is the product of the two individual probabilities;
1/6 * 1/6 = 1/36 (2.78%)
This means that it is less likely to roll the correct combination than it is to roll each of the numbers individually, which should intuitively make sense.
However, because there are an equal number of outcomes in which each player wins, this game is fair to both players. It is no more likely that the sum of the dice will be 9 than it is that it will be 9, even though both of these outcomes have relatively low probabilities. In most rolls, neither player will win - but the game is still fair to both players.