To get a probability in a given set, we need to count the events we want to happen and divide by the total possibilities.
a) Here, we have a set that goes from 1 to 12, so there is 12 possibilities. We want to pick a prime number, so we need to count how many primes we have in this set.
1 is not prime.
Also, 4, 6, 8, 9, 10 and 12 are not primes.
So, we have the primes: 2, 3, 5, 7 and 11. There are 5.
So, the probability will be:
[tex]P=\frac{5}{12}\approx0.42[/tex]b) Assuming the die are 6-sided going from 1 to 6, we can obtain the numbers from 1 + 1 = 2 until 6 + 6 = 12. However, there are differento number of possibilities. We still are looking for 2, 3, 5, 7 and 11, however now we have a total of 6 times 6 possibilities:
[tex]C_T=6\cdot6=36[/tex]And we have to calculate the combinations for each prime and add them.
2: there is only 1 + 1, so:
[tex]C_2=1[/tex]3: We can do 1 + 2 and 2 + 1, so there are 2:
[tex]C_3=2[/tex]5: We have 1 + 4, 2 + 3, 3 + 2 and 4 + 1, so 4 possibilities:
[tex]C_5=4[/tex]7: We have 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2 and 6 + 1, 6 possibilities:
[tex]C_7=6[/tex]11: We have 5 + 6 and 6 + 5 only. 2 possitilities:
[tex]C_{11}=2[/tex]In total, we have:
[tex]C_2+C_3+C_5+C_7+C_{11}=1+2+4+6+2=15_{}[/tex]So, the probability will be:
[tex]P=\frac{15}{36}\approx0.42[/tex]It ended up being the same.