Answer:
Simplifying this expression, we get P(A|B) = (1/5) * (36/11) = 36/55.
Therefore, the probability of event A occurring given that event B has occurred is 36/55.
Explanation:
To calculate P(A|B), which represents the probability of event A occurring given that event B has occurred, we need to understand the probability of each event individually.
Let's break down the problem step-by-step:
1. Event A: Rolling a three or four first, followed by an even number.
- To calculate the probability of rolling a three or four first, we need to find the probability of rolling a three or four on the first roll. Since there are two favorable outcomes (rolling a three or four) out of six possible outcomes (rolling numbers one to six), the probability of rolling a three or four on the first roll is 2/6 or 1/3.
- To calculate the probability of rolling an even number after rolling a three or four, we need to find the probability of rolling an even number on the second roll. Out of the remaining five possible outcomes (rolling numbers one to six except for three and four), there are three favorable outcomes (rolling an even number). Therefore, the probability of rolling an even number on the second roll is 3/5.
To find the probability of event A, we need to multiply the probabilities of each step: P(A) = (1/3) * (3/5) = 1/5.
2. Event B: The sum of the two rolls is at most seven.
- To calculate the probability of the sum of the two rolls being at most seven, we need to count the favorable outcomes where the sum is six or less. There are eleven favorable outcomes: (1,1), (1,2), (1,3), (1,4), (1,5), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2).
- Since there are a total of thirty-six possible outcomes when rolling two dice (six possible outcomes on the first roll and six possible outcomes on the second roll), the probability of event B is 11/36.
Now, we can calculate P(A|B) by dividing the probability of event A and event B occurring together by the probability of event B: P(A|B) = (1/5) / (11/36).
Simplifying this expression, we get P(A|B) = (1/5) * (36/11) = 36/55.
Therefore, the probability of event A occurring given that event B has occurred is 36/55.