Question

chapter 4.3 homework
score: 2/42 answered: 2/33
question 3
4.3 conditional probability. calculating conditional probabilities.
suppose you roll 2 dice. what is the pobability that the sum of the 2 dice is ≤ 4, given that the 1st dice is a 3?
express your answer as a fraction.

Explanation:

Step1: Determine the sample - space given the first die is 3

If the first die is 3, the possible outcomes for the two - die roll are (3,1), (3,2), (3,3), (3,4), (3,5), (3,6). So there are 6 possible outcomes.

Step2: Find the favorable outcomes

We want the sum of the two dice to be ≤ 4. If the first die is 3, then for the sum of the two dice \(x + y\leq4\) (where \(x = 3\)), when \(y = 1\), the sum \(x + y=3 + 1=4\). There is 1 favorable outcome.

Step3: Calculate the conditional probability

The formula for conditional probability \(P(A|B)=\frac{n(A\cap B)}{n(B)}\). Here, \(n(B) = 6\) (total number of outcomes when the first die is 3) and \(n(A\cap B)=1\) (number of outcomes where the first die is 3 and the sum is ≤ 4). So the probability is \(\frac{1}{6}\).

Answer:

\(\frac{1}{6}\)