Question 1199787: You roll two six-sided fair dice.
a. Let A be the event that either a 4 or 5 is rolled first followed by an even number.
P(A) =
Round your answer to four decimal places.
b. Let B be the event that the sum of the two dice is at most 5.
P(B) =
Round your answer to four decimal places.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part (a)
Define these two events:
C = event of rolling a 4 or 5 on a single die
D = event of rolling an even number on a single die
Each roll of the die is independent.
P(C) = 2/6 = 1/3
P(D) = 3/6 = 1/2
P(A) = P(C and D)
P(C and D) = P(C)*P(D) ... independent events
P(C and D) = (1/3)*(1/2)
P(C and D) = 1/6
P(A) = 1/6 = 0.1667
The 6's go on forever in the decimal value, but we round to four decimal places.
Answer: 0.1667
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Part (b)
B = event of the sum is at most 5
B = event the sum is 5 or smaller
B = event of the sum being in the set {2,3,4,5}
Sum of Dice Chart
| + | 1 | 2 | 3 | 4 | 5 | 6 | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | 3 | 4 | 5 | 6 | 7 | 8 | 9 | | 4 | 5 | 6 | 7 | 8 | 9 | 10 | | 5 | 6 | 7 | 8 | 9 | 10 | 11 | | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Example calculation: rolling 1 on the blue die, and rolling a 6 on the red die gives a sum of 1+6 = 7 (upper right corner)
The goal is to get a sum of 5 or smaller.
Focus on the following subset of sums from that chart
1+1 = 2
1+2 = 3
1+3 = 4
1+4 = 5
............
2+1 = 3
2+2 = 4
2+3 = 5
............
3+1 = 4
3+2 = 5
............
4+1 = 5
Take note how I'm breaking up the calculations.
The first group involves "1" as the first value, the second group involves "2" as the first value, and so on.
This allows for a predictable pattern and structure.
There are 4 items in the first group, 3 in the second, 2 in the third, and 1 in the fourth.
4+3+2+1 = 10 sums in which belong to the set {2,3,4,5}
There are 10 instances of achieving event B out of 6*6 = 36 ways to roll two dice.
10/36 = (2*5)/(2*18) = 5/18 is the probability that event B occurs.
Use a calculator or long division to find that:
5/18 = 0.2778
The 7's go on forever, but we round to four decimal places.
Answer: 0.2778
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