To solve these probability problems, we can use the concept of combinations and the binomial probability formula.
For question 11:
The probability of guessing the correct answer to a multiple-choice question with 5 options is 1/5 or 0.2. Since you didn't study and are guessing all 20 questions, the probability of getting a correct answer on any given question is 0.2.
The probability of guessing at least 50% of the problems correctly can be calculated using the binomial probability formula:
P(X ≥ k) = 1 - P(X < k)
Here, X follows a binomial distribution with n = 20 (number of trials/questions) and p = 0.2 (probability of success/guessing correctly).
Using this formula, we can calculate the probability:
P(X ≥ 50%) = 1 - P(X < 50%)
To calculate P(X < 50%), we need to sum up the probabilities of getting 0, 1, 2, ..., 9, 10 correct answers and subtract it from 1.
The calculation involves finding the probability of getting k correct answers as:
P(X = k) = C(n, k) * (p^k) * ((1-p)^(n-k))
Where C(n, k) represents the number of combinations of n items taken k at a time.
Summing up these probabilities for k = 0 to 10 and subtracting from 1 will give us the probability of getting at least 50% of the problems correct.
For question 12:
The probability that a fatal car accident involves drunk driving is 1/2 or 0.5. The probability of at least 2 out of 5 accidents being caused by drunk driving can be calculated using the binomial probability formula (similar to question 11), where n = 5 (number of accidents) and p = 0.5.
For question 13:
The probability that a fatal car accident involves drunk driving is 40% or 0.4. Using the same binomial probability formula as above, we can calculate the probability of at least one accident being caused by drunk driving with n = 5 (number of accidents) and p = 0.4.
For question 14:
Each power source for the satellite system works independently with a probability of 0.90. The satellite system will function if at least 2 out of 5 power sources work properly. We can calculate the probability of the satellite system functioning properly using the binomial probability formula, where n = 5 (number of power sources) and p = 0.90.
Now that we have the formulas and probabilities for each question, let's calculate the answers.