Two standard six-sided dice are rolled and the sum of their squares computed. Find the probability that the resulting number will be an element of the set
{2,5,8,10,13,17,18,20,26,29,32,34,37,41,45,52,61,72}
{2,5,8,10,13,17,18,20,26,29,32,34,37,41,45,52,61,72}
.
well, just list the possible outcomes:
1+1=2
1+4=5
1+9=10
...
16+36=52
25+36=61
36+36=72
how many outcomes?
how many set elements?
divide
To find the probability that the resulting number will be an element of a given set, we need to first determine the total number of possible outcomes when rolling two six-sided dice.
Each dice has 6 possible outcomes, resulting in a total of 6 x 6 = 36 possible outcomes when rolling two dice.
Next, we need to find the number of outcomes where the sum of the squares of the numbers rolled is part of the given set.
We can calculate the sum of the squares for each possible outcome and count how many times it matches with an element in the given set.
Here's the breakdown of the sum of squares for all possible outcomes:
2: Only one outcome: (1, 1)
5: Two outcomes: (1, 2) and (2, 1)
8: Three outcomes: (2, 2), (2, 4), and (4, 2)
10: Three outcomes: (1, 3), (3, 1), and (2, 4)
13: Four outcomes: (2, 3), (3, 2), (4, 1), and (1, 4)
17: One outcome: (4, 3)
18: Three outcomes: (3, 3), (3, 4), and (4, 3)
20: Two outcomes: (2, 5) and (5, 2)
26: One outcome: (5, 3)
29: One outcome: (4, 5)
32: One outcome: (4, 4)
34: Two outcomes: (3, 5) and (5, 3)
37: One outcome: (5, 4)
41: Two outcomes: (4, 5) and (5, 4)
45: Four outcomes: (4, 6), (6, 4), (5, 5), and (5, 6)
52: One outcome: (6, 4)
61: One outcome: (5, 6)
72: One outcome: (6, 6)
In total, there are 23 outcomes where the sum of the squares matches with an element in the given set.
Finally, to find the probability, we divide the number of desired outcomes (23) by the total number of possible outcomes (36):
P = 23/36
Therefore, the probability that the resulting number will be an element of the set {2,5,8,10,13,17,18,20,26,29,32,34,37,41,45,52,61,72} is 23/36.