Question

two number cubes are rolled for two separate events event A is the event that the sum of the numbers on both cubes is less than 10 event B is the event that the sum of the numbers on both cubes is a multiple of 3 find the conditional probability of B given A occurs first enter your answer as a simplified fraction

Answer 1

We have two events:

A: sum of the numbers is less than 10.

B: sum of the numbers is a multiple of 3.

We can calculate the probabilities as a quotient of the "success" events and all the possible events.

The conditional probability P(B | A) is equal to the probability of P(A intersection B) divided by P(A). That is because, if A is given, then if B happens, A had also happen.

The "success" events for intersection A and B are:

{1,2}, {2,1}, {1,5}, {5,1}, {2,4}, {4,2}, {3,3}, {3,6}, {6,3}, {4,5}, {5,4}

There are a total of 11 results that belong to the intersection of A and B (sum less than 10 and multiples of 3).

Now, we calculate the results that correspond to event A:

{1,1}, {1,2}, {2,1}, {1,3}, {3,1}, {1,4}, {4,1}, {1,5}, {5,1}, {1,6}, {6,1}

{2,2}, {2,3}, {3,2}, {2,4}, {4,2}, {2,5}, {5,2}, {2,6}, {6,2}

{3,3}, {3,4}, {4,3}, {3,5}, {5,3}, {3,6}, {6,3}

{4,4}, {4,5}, {5,4}

There are 30 results that correspond to event A (sum is less than 10).

Then we can calculate P(B | A) as:

`P(B|A)=(P(A\cap B))/(P(A))=(11)/(30)`

The conditional probability of B given A is P(B|A) = 11/30.