The probability that a student will absent from class is 0.04, assuming that the student was in attendance the previous school day. However if a student is absent the probability that he or she will be absent the next day is 0.11. Assume that the student was in attendance the previous day. What is the probability that the student will be absent 2 days in a row but will show up the third day?

To solve this problem, we need to calculate the probability that the student will be absent two days in a row and then attend class on the third day.

Let's break down the problem step by step:

1. **Day 1 (Given that the student attended the previous day):**
   The probability that the student will be absent on the first day (denoted as \( A_1 \)) is given as 0.04.

   \[
   P(A_1) = 0.04
   \]

2. **Day 2 (Given that the student was absent on the first day):**
   The probability that the student will be absent again on the second day (denoted as \( A_2 \)) is 0.11, given that they were absent the previous day.

   \[
   P(A_2 \mid A_1) = 0.11
   \]

3. **Day 3 (Given that the student was absent on both the first and second days):**
   The probability that the student will attend on the third day (denoted as \( T_3 \)) is the complement of the probability of being absent on the third day. That is, the probability of attending is \( 1 - 0.11 = 0.89 \), given that they were absent the previous two days.

   \[
   P(T_3 \mid A_2) = 0.89
   \]

### Overall Probability:

The total probability that the student will be absent for two consecutive days and attend on the third day is the product of the conditional probabilities:

\[
P(A_1 \cap A_2 \cap T_3) = P(A_1) \times P(A_2 \mid A_1) \times P(T_3 \mid A_2)
\]

Substituting the known values:

\[
P(A_1 \cap A_2 \cap T_3) = 0.04 \times 0.11 \times 0.89
\]

\[
P(A_1 \cap A_2 \cap T_3) = 0.003916
\]

Thus, the probability that the student will be absent for two consecutive days and then attend on the third day is **0.003916** or approximately **0.39%**.

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Would you like further details or clarification?

Here are 5 related questions that expand on this topic:

1. What is the probability that the student will be absent for exactly one day?
2. How would the probability change if the student’s attendance pattern on a third day was different?
3. What is the probability that the student will attend all three days?
4. How would this scenario change if the initial attendance probability on day 1 was different?
5. How does this problem relate to Markov chains in probability theory?

**Tip:** When dealing with conditional probabilities, always check if the problem assumes dependence or independence between events!

Explore how to calculate the probability of a student being absent for two consecutive days and then attending class on the third day. This problem applies conditional probability and the multiplication rule for dependent events, suitable for Grades 9-12.

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