Question 1178069: You and your friend are taking the driving test to obtain your driver’s license. The probability for you to pass the test is 0.9 and for your friend, it is 0.8. Calculate the probability that you two make a total of 6 attempts to obtain your licenses. (if you got your license on the 2nd attempt and your friend got his on 3rd attempt, you made a total of 5 attempts, for example)
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step.
**Understanding the Problem**
* **Your Probability of Passing:** 0.9 (Probability of Failure: 0.1)
* **Friend's Probability of Passing:** 0.8 (Probability of Failure: 0.2)
* **Total Attempts:** 6
* **Goal:** Find the probability that the total number of attempts you and your friend make is exactly 6.
**Possible Scenarios**
Let's denote your attempts as 'Y' and your friend's attempts as 'F'. We need to find all pairs (Y, F) such that Y + F = 6.
* (1, 5)
* (2, 4)
* (3, 3)
* (4, 2)
* (5, 1)
**Calculating Probabilities for Each Scenario**
* **Y = 1:** You pass on the first attempt. Probability = 0.9
* **Y = 2:** You fail once, then pass. Probability = (0.1)(0.9)
* **Y = 3:** You fail twice, then pass. Probability = (0.1)(0.1)(0.9)
* **Y = 4:** You fail three times, then pass. Probability = (0.1)(0.1)(0.1)(0.9)
* **Y = 5:** You fail four times, then pass. Probability = (0.1)(0.1)(0.1)(0.1)(0.9)
* **F = 1:** Your friend passes on the first attempt. Probability = 0.8
* **F = 2:** Your friend fails once, then passes. Probability = (0.2)(0.8)
* **F = 3:** Your friend fails twice, then passes. Probability = (0.2)(0.2)(0.8)
* **F = 4:** Your friend fails three times, then passes. Probability = (0.2)(0.2)(0.2)(0.8)
* **F = 5:** Your friend fails four times, then passes. Probability = (0.2)(0.2)(0.2)(0.2)(0.8)
**Calculating Probabilities for Each Pair (Y, F)**
* **(1, 5):** (0.9) * (0.2)^4 * (0.8) = 0.9 * 0.0016 * 0.8 = 0.001152
* **(2, 4):** (0.1)(0.9) * (0.2)^3 * (0.8) = 0.09 * 0.008 * 0.8 = 0.000576
* **(3, 3):** (0.1)^2 * (0.9) * (0.2)^2 * (0.8) = 0.009 * 0.04 * 0.8 = 0.000288
* **(4, 2):** (0.1)^3 * (0.9) * (0.2) * (0.8) = 0.0009 * 0.16 = 0.000144
* **(5, 1):** (0.1)^4 * (0.9) * (0.8) = 0.00009 * 0.8 = 0.000072
**Summing the Probabilities**
Total Probability = 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.**
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