Answer:
(a) The probability of getting all sixes is 1/7776.
(b) The probability of getting all the same outcomes is 1/1296.
(c) The probability of getting all different outcomes is 5/54.
Explanation:
(a) The probability of getting all sixes when rolling five dice can be calculated by determining the probability of rolling a six on one die and then raising it to the power of five since there are five dice.
The probability of rolling a six on one die is 1/6, since there are six possible outcomes (numbers 1 to 6) and only one of them is a six.
Therefore, the probability of getting all sixes when rolling five dice is (1/6)^5, which simplifies to 1/7776.
(b) The probability of getting all the same outcomes when rolling five dice can be calculated by considering the number of favorable outcomes (when all the dice show the same number) divided by the total number of possible outcomes.
There are six possible outcomes (numbers 1 to 6) that can be rolled on the first die. For each outcome on the first die, there is only one outcome on each subsequent die that would make all the dice show the same number.
Therefore, the probability of getting all the same outcomes when rolling five dice is 1/6 * 1/6 * 1/6 * 1/6, which simplifies to 1/1296.
(c) The probability of getting all different outcomes when rolling five dice can be calculated by considering the number of favorable outcomes (when each die shows a different number) divided by the total number of possible outcomes.
The first die can show any number from 1 to 6, giving us six possible outcomes. For each outcome on the first die, there are five remaining numbers that can be rolled on the second die, four remaining numbers for the third die, three for the fourth die, and two for the fifth die.
Therefore, the probability of getting all different outcomes when rolling five dice is 6/6 * 5/6 * 4/6 * 3/6 * 2/6, which simplifies to 5/54.