How do you calculate the probability of a sum with one fair die and one biased die?
P(1)=1/3
P(2)=P(3)=P(4)=P(5)=P(6)
I presume you gave the probability of the biased die, where there is a 33.33% chance of rolling a 1 and a 13.33% chance of rolling a 2,3,4,5 or 6. For a normal die, the probability of rolling any given number is 1/6 = 16.67%.
There is one way to roll a 2, both dies are 1. so, Psum(2) = .1666 * .3333 = .0555
There are two ways to roll an 3, 1-2, and 2-1. So Psum(3) = (.16666*.3333)+(.16666*.13333) = .0777
Repeat for values 4 to 12.
To calculate the probability of a sum with one fair die and one biased die, you first need to determine the probabilities for each outcome when rolling each die separately.
Let's consider the probabilities for the fair die first. Since it is fair, each outcome (1, 2, 3, 4, 5, 6) has an equal chance of occurring. Therefore, the probability for each outcome is 1/6.
Now, let's consider the probabilities for the biased die. According to the given information, the probabilities for the outcomes 1, 2, 3, 4, 5, 6 are different. Let's denote these probabilities as P(1), P(2), P(3), P(4), P(5), P(6) respectively.
To calculate the probability of the sum, you need to find the probability of each combination of outcomes that would result in the desired sum. For example, to find the probability of getting a sum of 2, you need to determine all the pairs of outcomes (x, y) such that x + y = 2.
To get a sum of 2, there is only one possible combination: (1, 1). Therefore, the probability of getting a sum of 2 is P(1) * (1/6).
To calculate the probability of the sum for each possible value, you can follow these steps:
- For each sum value, determine all the possible combinations of outcomes that result in that sum.
- Multiply the corresponding probabilities for each combination and sum them up.
For example, to calculate the probability of getting a sum of 3:
- The combinations that result in a sum of 3 are (1, 2), (2, 1).
- The probability of (1, 2) is P(1) * (1/6).
- The probability of (2, 1) is P(2) * (1/6).
- Add up these probabilities to get the final probability for a sum of 3.
You can repeat this process for each possible sum value to calculate its probability.