Consider an experiment in which a fair four-sided die (with faces labeled 0 , 1 , 2 , 3 ) is rolled once to determine how many times a six-sided die (with faces labeled 1 ,.., 6 ) is rolled. The die rolls are independent. Let X be the total number of times the six-sided die results in 5 or 6 .
1. Calculate the conditional probability that X=2 given that the four-sided die roll resulted in 2.
2. Calculate P(X=2) .
3. Given that X=2 , what is the probability that the result of the four-sided die roll was 3?
1. The conditional probability that X=2 given that the four-sided die roll resulted in 2 is 1/3.
2. The probability that X=2 is 1/4.
3. Given that X=2 , the probability that the result of the four-sided die roll was 3 is 1/3.
1. To calculate the conditional probability that X=2 given that the four-sided die roll resulted in 2, we need to use Bayes' theorem.
Bayes' theorem states that P(A|B) = (P(B|A) * P(A)) / P(B), where A and B are two events. In this case, event A is X=2 (the total number of times the six-sided die results in 5 or 6), and event B is the four-sided die roll resulting in 2.
To find P(X=2 | four-sided roll = 2), we need to find the following probabilities:
P(X=2) is the probability that X equals 2, which we will calculate in step 2.
P(four-sided roll = 2 | X=2) is the probability that the four-sided die roll resulted in 2, given that X=2. Since the four-sided die roll and the six-sided die rolls are independent, this probability is simply 1/4.
P(four-sided roll = 2) is the probability that the four-sided die roll resulted in 2, which is also 1/4 since the die is fair.
Now, we can apply Bayes' theorem:
P(X=2 | four-sided roll = 2) = (P(four-sided roll = 2 | X=2) * P(X=2)) / P(four-sided roll = 2)
P(X=2 | four-sided roll = 2) = (1/4 * P(X=2)) / (1/4)
The probability of the four-sided die roll resulting in 2 cancels out, leaving us with:
P(X=2 | four-sided roll = 2) = P(X=2)
Therefore, the conditional probability that X=2 given that the four-sided die roll resulted in 2 is equal to P(X=2).
2. To calculate P(X=2), we need to consider all possible outcomes of the experiment and determine the probability of each outcome.
There are two parts to this experiment: the four-sided die roll and the subsequent six-sided die rolls. We need to consider all possible combinations of these rolls that result in X=2.
If the four-sided die roll results in 0 or 1, X will be 0 since there will be no opportunities for the six-sided die to roll 5 or 6.
If the four-sided die roll results in 2, there will be one opportunity for the six-sided die to roll 5 or 6. Since the six-sided die is fair, the probability of this happening is 2/6 = 1/3.
If the four-sided die roll results in 3, there will be two opportunities for the six-sided die to roll 5 or 6. The probability of the first roll being 5 or 6 is 2/6 = 1/3, and the probability of the second roll being 5 or 6 is also 1/3. Therefore, the probability of this outcome is (1/3) * (1/3) = 1/9.
The probability of X=2 is the sum of these probabilities:
P(X=2) = P(four-sided roll = 2) * P(one opportunity for six-sided roll) + P(four-sided roll = 3) * P(two opportunities for six-sided roll)
P(X=2) = (1/4) * (1/3) + (1/4) * (1/9)
P(X=2) = 1/12 + 1/36
P(X=2) = 3/36 + 1/36
P(X=2) = 4/36
Simplifying, P(X=2) = 1/9.
Therefore, the probability that X equals 2 is 1/9.
3. Given that X=2, we want to find the probability that the result of the four-sided die roll was 3.
To calculate this probability, we need to use Bayes' theorem again:
P(four-sided roll = 3 | X=2) = (P(X=2 | four-sided roll = 3) * P(four-sided roll = 3)) / P(X=2)
P(X=2 | four-sided roll = 3) = 2/6 = 1/3, as there are two opportunities for the six-sided die to roll 5 or 6 if the four-sided die roll is 3.
P(four-sided roll = 3) = 1/4, the probability of the four-sided die roll resulting in 3, which is also 1/4 since the die is fair.
P(X=2) = 1/9, as calculated in step 2.
Now we can use Bayes' theorem:
P(four-sided roll = 3 | X=2) = (1/3 * 1/4) / (1/9)
P(four-sided roll = 3 | X=2) = 1/12 / 1/9
P(four-sided roll = 3 | X=2) = 1/12 * 9/1
P(four-sided roll = 3 | X=2) = 9/12
Simplifying, P(four-sided roll = 3 | X=2) = 3/4.
Therefore, given that X=2, the probability that the result of the four-sided die roll was 3 is 3/4.