Question 1177708
Let's break down this problem.

**Understanding the Problem**

* You: Probability of passing (P_you) = 0.9, Probability of failing (F_you) = 0.1
* Friend: Probability of passing (P_friend) = 0.8, Probability of failing (F_friend) = 0.2
* We want the probability that the total number of attempts is 6.

**Possible Scenarios**

To have a total of 6 attempts, we need to consider all combinations of attempts that sum to 6. These are:

* You: 1 attempt, Friend: 5 attempts
* You: 2 attempts, Friend: 4 attempts
* You: 3 attempts, Friend: 3 attempts
* You: 4 attempts, Friend: 2 attempts
* You: 5 attempts, Friend: 1 attempt

**Calculating Probabilities**

We'll use the geometric distribution to find the probability of passing on a specific attempt. The probability of passing on the nth attempt is:

* P(n attempts) = (failure probability)^(n-1) * (success probability)

Let's calculate the probabilities for each scenario:

1.  **You: 1 attempt, Friend: 5 attempts**
    * P(You = 1) = 0.9
    * P(Friend = 5) = (0.2)^4 * 0.8 = 0.00128
    * Probability of this scenario: 0.9 * 0.00128 = 0.001152

2.  **You: 2 attempts, Friend: 4 attempts**
    * P(You = 2) = 0.1 * 0.9 = 0.09
    * P(Friend = 4) = (0.2)^3 * 0.8 = 0.0064
    * Probability of this scenario: 0.09 * 0.0064 = 0.000576

3.  **You: 3 attempts, Friend: 3 attempts**
    * P(You = 3) = (0.1)^2 * 0.9 = 0.009
    * P(Friend = 3) = (0.2)^2 * 0.8 = 0.032
    * Probability of this scenario: 0.009 * 0.032 = 0.000288

4.  **You: 4 attempts, Friend: 2 attempts**
    * P(You = 4) = (0.1)^3 * 0.9 = 0.0009
    * P(Friend = 2) = 0.2 * 0.8 = 0.16
    * Probability of this scenario: 0.0009 * 0.16 = 0.000144

5.  **You: 5 attempts, Friend: 1 attempt**
    * P(You = 5) = (0.1)^4 * 0.9 = 0.00009
    * P(Friend = 1) = 0.8
    * Probability of this scenario: 0.00009 * 0.8 = 0.000072

**Total Probability**

Add the probabilities of each scenario:

* 0.001152 + 0.000576 + 0.000288 + 0.000144 + 0.000072 = 0.002232

**Therefore, the probability that you two make a total of 6 attempts to obtain your licenses is 0.002232.**