f two indistinguishable dice are rolled, what is the probability of the event {(3, 3), (2, 3), (5, 3)}?
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What is the corresponding event for a pair of distinguishable dice?
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{(3, 3), (2, 3), (5, 3)} {(3, 3), (2, 3), (5, 3), (3, 5), (3, 2)} {(3, 3), (2, 2), (5, 5), (3, 5), (3, 2)} {(2, 3), (5, 3), (3, 5), (3, 2)} {(2, 3), (5, 3)}
total no. of the combination of two dice = 21
3/21=1/7
To find the probability of an event, we need to divide the number of favorable outcomes by the total number of possible outcomes.
For a pair of indistinguishable dice, we have 6 possible outcomes for each roll. Therefore, the total number of possible outcomes for two rolls is 6 * 6 = 36.
The favorable outcomes for the event {(3, 3), (2, 3), (5, 3)} are (3, 3), (2, 3), and (5, 3), which means 3 outcomes.
So, the probability of the event {(3, 3), (2, 3), (5, 3)} for indistinguishable dice is 3/36 or 1/12.
For a pair of distinguishable dice, we have 6 possible outcomes for the first die and 6 possible outcomes for the second die, resulting in a total of 6 * 6 = 36 possible outcomes.
The corresponding event for distinguishable dice would include all the outcomes from the indistinguishable dice event, plus additional outcomes where the order matters. So the corresponding event would be {(3, 3), (2, 3), (5, 3), (3, 5), (3, 2)}. This event has 5 outcomes.
Thus, the probability of the corresponding event for distinguishable dice is 5/36.
To find the probability of an event, we need to divide the number of favorable outcomes by the total number of possible outcomes.
In this case, we have two indistinguishable dice rolled, so each dice has 6 possible outcomes (1, 2, 3, 4, 5, 6).
The event {(3, 3), (2, 3), (5, 3)} has 3 favorable outcomes. We can count them: (3, 3), (2, 3), and (5, 3).
So, the probability of this event is 3 favorable outcomes out of 36 total possible outcomes.
Therefore, the probability is 3/36, which can be simplified to 1/12.
For the corresponding event for a pair of distinguishable dice, we need to consider all possible outcomes where the order matters.
In this case, the corresponding event would be {(3, 3), (2, 3), (5, 3), (3, 5), (3, 2)}. We can also write it as {(3, 3), (2, 3), (5, 3), (3, 5), (3, 2)}.
This event includes all the favorable outcomes, where both dice show a 3, as well as the outcomes where only one of the dice shows a 3.
It's important to note that {(3, 3), (2, 2), (5, 5)} would not be the corresponding event because it only takes into account the favorable outcome where both dice show the same value.
So, the corresponding event for a pair of distinguishable dice would be {(3, 3), (2, 3), (5, 3), (3, 5), (3, 2)}.