Answer :
Step-by-step explanation:
there are 36 (6×6) possible results and number combinations for 2 dice.
a)
to create a sum of 4 we need one of the following combinations :
1 3
2 2
3 1
so, 3 out of the overall possible 36 possibilities.
and that means the probability is 3/36 = 1/12
b)
to create a sum of 7 we need one of the following combinations :
1 6
2 5
3 4
4 3
5 2
6 1
so, 6 out of the overall possible 36 possibilities.
and that means a probability of 6/36 = 1/6
c)
since the possible outcomes in that regard are only 2 (even and uneven), with equal balance the probability is 1/2.
make it more formal ?
to get an even sum, every possible number on one die can be combined with 3 possibilities on the other.
1 can be combined with 1, 3, 5
2 with 2, 4, 6
3 with 1, 3, 5
4 with 2, 4, 6
5 with 1, 3, 5
6 with 2, 4, 6
these are 6×3 = 18 combinations of of the possible 36.
so, the probability is 18/36 = 1/2
d)
not a 6 ?
so, what does that mean ?
we are looking only at 5 possible outcomes per die.
that is 5×5 = 25 combinations out of the possible 36.
that means that probabilty is 25/36.
e)
not a perfect square ?
we have in the range of the possible sums with 2 dice (2 .. 12) only two perfect squares : 4 and 9
so, the sum must NOT be 4 AND NOT be 9.
so, here it might be easier to count the unwanted cases and then deduct the probability from 1 to express the opposite.
the combinations to get 4 we have already under a)
3 combinations with 3/36=1/12 probability.
the combinations to get 9
1 none
2 none
3 6
4 5
5 4
6 3
so, 4 combinations with a probability of 4/36 = 1/9
therefore, we have (3+4) = 7 cases out of the possible 36 we actually want to avoid.
so, the probability to get any one of the other (desired) combinations is therefore (36-7)/36 = 29/36
The probabilities are:
- a) 0.083
- b) 0.167
- c) 0.5
- d) 0.861
- e) 0.806.
How to get the probabilities?
The probability for a given outcome is given by the quotient between the number of combinations that give that outcome and the total number of combinations.
Here each dice has 6 possible outcomes, so the total number of combinations for the two dice is:
6*6 = 36 combinations.
a) Here we need to count how many combinations add to 4, the notation I will use is:
dice 1 dice 2
1 3
3 1
2 2
There are 3 combinations that add to 4, then the probability of getting a sum equal to 4 is:
P = 3/36 = 0.083
b) Same thing as before, this time we count the combinations that add to 7.
dice 1 dice 2
1 6
6 1
5 2
2 5
4 3
3 4
There are 6 combinations that add to 7, so the probability is:
P = 6/36 = 0.167
c) The possible even numbers are: {2, 4, 6, 8, 10, 12}
A way of doing this, is:
If the first dice is even/odd, then the second also must be even/odd.
Then the probability is just 1/2 = 0.5 (because there are the same number of odd and even numbers in a dice).
d) The sum must not be equal to 6, the combinations that add up to 6 are:
dice 1 dice 2
2 4
4 2
3 3
1 5
5 1
So 5 combinations add up to 6, then there are 31 combinations that do not add up to 6, this means that the probability of not getting a sum equal to 6 is:
P = 31/36 = 0.861
e) The perfect squares are 4 and 9, we already know that 3 combinations add up to 4, now let's find the combinations that add up to 9.
dice 1 dice 2
6 3
3 6
5 4
4 5
So 4 combinations add up to 9, then there is a total of 7 combinations that give a perfect square, then there are 29 combinations that do not add to a perfect square.
The probability is:
P = 29/36 = 0.806
If you want to learn more about probability, you can read:
https://brainly.com/question/251701