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Two dice are rolled. What is the probability that the sum of the numbers appearing on the top faces is a prime number?
5/12
1/2
7/18
13/36
5/12
Memorize the distribution of sums for two dice: 2(1), 3(2), 4(3), 5(4), 6(5), 7(6), 8(5), 9(4), 10(3), 11(2), 12(1). Sum the counts for primes {2, 3, 5, 7, 11}: 1+2+4+6+2=15. Divide by 36 to get 15/36=5/12.
Two fair six-sided dice are rolled simultaneously.
P(E)=Total┬аnumber┬аof┬аoutcomesNumber┬аof┬аfavorable┬аoutcomesтАЛ
Memorize the distribution of sums for two dice: 2(1), 3(2), 4(3), 5(4), 6(5), 7(6), 8(5), 9(4), 10(3), 11(2), 12(1). Sum the counts for primes {2, 3, 5, 7, 11}: 1+2+4+6+2=15. Divide by 36 to get 15/36=5/12.
Students often include 1 as a prime number or forget that 9 is not prime, leading to an incorrect count of outcomes.
Calculate Total Outcomes
Since each die has 6 faces, rolling two dice results in a total of 6├Ч6=36 possible outcomes.
n(S)=36
Identify Prime Sums
The possible sums for two dice range from 2 to 12. The prime numbers in this range are 2, 3, 5, 7, and 11.
PrimesтИИ{2,3,5,7,11}
Count Favorable Outcomes
Sum = 2: (1,1); Sum = 3: (1,2), (2,1); Sum = 5: (1,4), (4,1), (2,3), (3,2); Sum = 7: (1,6), (6,1), (2,5), (5,2), (3,4), (4,3); Sum = 11: (5,6), (6,5). Total favorable outcomes = 1+2+4+6+2=15.
n(E)=15
Calculate Probability
Applying the probability formula: P(E)=15/36. Simplifying by dividing numerator and denominator by 3, we get 5/12.
P(E)=3615тАЛ=125тАЛ
A is correct because the number of favorable prime-sum outcomes is 15 out of 36 total possibilities, which simplifies to 5/12.
This concept of 'sum distributions' is vital in statistics for understanding discrete probability distributions and combinatorics.