Question 1177708: 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.
**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.**
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