3. Yahtzee is a game in which a player rolls five six-sided dice simultaneously. The actual...

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3. Yahtzee is a game in which a player rolls five six-sided dice simultaneously. The actual rules are a bit more involved, but for the sake of simplicity, let us only consider the case of rolling all five dice at once. (a) What is the probability of obtaining all unique outcomes on each of the five dice rolled? For example, {1,2,4,5,6} is all unique outcomes, but {1,2,4,5,2} repeats a 2, so those outcomes are not all unique. (b) A "small straight" consists of four numbers all in a sequence, plus one other die that can be anything. For example, one such outcome could be {2, 3, 4, 5, 3), and another could be {2, 3, 4, 5, 6} (we are ignoring the distinction between a small and a large straight). What is the probability of rolling a small straight when you roll all five dice? Caution: beware of counting specific outcomes multiple times! (c) You roll all five dice but then your phone alerts you that you've caught a new Pokémon with your Fortnite. Nice! You become so engrossed in the video games that you don't see what the outcome of the dice rolls is. Your kind friend tells you that the dice are all unique. Given this information, what then is the probability of that you have rolled a small straight? (d) Are the events [roll a small straight] and [roll all unique outcomes] independent? Fully justify your answer with math. 3. Yahtzee is a game in which a player rolls five six-sided dice simultaneously. The actual rules are a bit more involved, but for the sake of simplicity, let us only consider the case of rolling all five dice at once. (a) What is the probability of obtaining all unique outcomes on each of the five dice rolled? For example, {1,2,4,5,6} is all unique outcomes, but {1,2,4,5,2} repeats a 2, so those outcomes are not all unique. (b) A "small straight" consists of four numbers all in a sequence, plus one other die that can be anything. For example, one such outcome could be {2, 3, 4, 5, 3), and another could be {2, 3, 4, 5, 6} (we are ignoring the distinction between a small and a large straight). What is the probability of rolling a small straight when you roll all five dice? Caution: beware of counting specific outcomes multiple times! (c) You roll all five dice but then your phone alerts you that you've caught a new Pokémon with your Fortnite. Nice! You become so engrossed in the video games that you don't see what the outcome of the dice rolls is. Your kind friend tells you that the dice are all unique. Given this information, what then is the probability of that you have rolled a small straight? (d) Are the events [roll a small straight] and [roll all unique outcomes] independent? Fully justify your answer with math.