two dice are rolled. if a 3 appears on either dice,then the two dice are rolled again and the two dice are added togather. if a 3 does not appear, then we ll add the two dice togather. if the sum of two dice is even ,i win and if the sum is odd,you win. what is the probability that you win? what is the probability that i winn/
It doesn't matter whether the 3 appears or not on the dice in the first roll. What matters is whether the sum of the winning roll is even or odd.
The sum will be odd if one dice is an even number, and one dice is an odd number
The sum will be even if both dice are even or both dice are odd.
The probability of any dice being even is 1/2
The probability of any dice being odd is 1/2
The probability of the first dice being even and the second dice being even is 1/2 * 1/2 = 1/4
The probability of the first dice being odd is 1/2 * 1/2 = 1/4
The probability of the first dice being even and the second being odd is 1/2*1/2 = 1/4
The probability of the first dice being odd and the second being even is 1/2*1/2 = 1/4
Therefore the probability of an even sum (even, even) or (odd,odd) is 1/4+ 1/4 = 1/2
And the probability of an odd sum (even, odd) or (odd, even) is 1/4+1/4 = 1/2
To find the probability of winning, we need to analyze the possible outcomes for each roll of the dice and count the favorable outcomes.
Let's break down the possible scenarios:
1. Rolling a 3 on either dice:
- In this case, we roll the two dice again, and the sum is added together to determine the outcome.
2. Not rolling a 3 on either dice:
- In this case, the sum of the two dice is considered as the outcome.
Let's calculate the probability of each scenario:
1. Rolling a 3 on either dice:
- Probability of rolling a 3 on a single die = 1/6
- Probability of not rolling a 3 on a single die = 5/6 (since there are 6 possible outcomes on a die)
- Probability of rolling a 3 on either dice = 1/6 + 1/6 = 1/3
2. Not rolling a 3 on either dice:
- Probability of not rolling a 3 on a single die = 5/6
- Probability of not rolling a 3 on both dice = (5/6) * (5/6) = 25/36
- Probability of rolling a sum that is even = probability of rolling a 2, 4, 6, 8, 10, or 12 (out of the possible sums of 2, 3, ..., 12)
- The sum of the probabilities for each even sum is (1/36 + 3/36 + 5/36 + 7/36 + 9/36 + 11/36) = 36/36 = 1
Now, let's find the probability of winning and losing:
Probability of winning = Probability of rolling a 3 on either dice * Probability of rolling an even sum given no 3's + Probability of not rolling a 3 on either dice * Probability of rolling an even sum.
Probability of winning = (1/3) * (1) + (2/3) * (1) = 1/3 + 2/3 = 3/3 = 1
Probability of losing = 1 - Probability of winning = 1 - 1 = 0
Therefore, the probability of you winning is 1, and the probability of you losing is 0.