A three-dice is a six sided dice with only the numbers 1, 2 and 3. The usual numbers 4, 5 and 6 are replaced by 1, 2 and 3 respectively.
4 three-dice (with sides labeled 1, 2, and 3) are rolled and the numbers face up on the 4 dice are recorded. Give the sample space for this experiment. What is the probability that at least one of the dice rolled is showing a 2?
What is the probability that at least one of the dice rolled is showing a 2?=
What is the probability that at least one of the dice rolled is showing a 3?=
To find the sample space and the probability for this experiment, we need to understand the basic concepts of sample space and probability.
Sample Space:
The sample space is the set of all possible outcomes of an experiment. In this case, we are rolling four three-dice, so each dice can show one of the three numbers: 1, 2, or 3. Therefore, the sample space is the set of all possible combinations of numbers that can be rolled on the four dice.
To find the sample space, we can list all the possible outcomes:
1,1,1,1
1,1,1,2
...
3,3,3,3
There are a total of 3^4 = 81 possible outcomes in the sample space.
Probability:
Probability is a measure of the likelihood of an event occurring. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
In this case, the event is that at least one of the dice rolled is showing a 2 or 3.
Probability that at least one of the dice rolled is showing a 2:
To calculate this probability, we need to count the number of favorable outcomes and divide it by the total number of outcomes in the sample space.
Let's find the number of favorable outcomes. There are four different ways in which at least one of the dice can show a 2:
1. Only the first dice shows a 2, and the other three dice show any number (3*3*3 = 27 outcomes).
2. Only the second dice shows a 2, and the other three dice show any number (3*3*3 = 27 outcomes).
3. Only the third dice shows a 2, and the other three dice show any number (3*3*3 = 27 outcomes).
4. Only the fourth dice shows a 2, and the other three dice show any number (3*3*3 = 27 outcomes).
However, we have counted the outcome where all four dice show a 2 four times. Therefore, we need to subtract this outcome once from the total.
So, the number of favorable outcomes is 27+27+27+27-1 = 107.
The probability that at least one of the dice rolled is showing a 2 is given by:
P(at least one 2) = Number of favorable outcomes / Total number of outcomes
= 107 / 81
= 1.32 (rounded to two decimal places)
Probability that at least one of the dice rolled is showing a 3:
The calculation for the probability that at least one of the dice rolled is showing a 3 follows a similar approach as above.
We again have four different ways in which at least one of the dice can show a 3, with each dice in the sample space having three possible outcomes.
So, the number of favorable outcomes is 27+27+27+27-1 = 107 (same as before).
The probability that at least one of the dice rolled is showing a 3 is given by:
P(at least one 3) = Number of favorable outcomes / Total number of outcomes
= 107 / 81
= 1.32 (rounded to two decimal places)
Therefore, the probability that at least one of the dice rolled is showing a 2 or 3 is 1.32 or 132%.