11. You are taking a multiple choice examination that has 20 questions in a particular section
with 5 possible answers for each question. You did not study for the test and you
guessed the answers. The probability that you will get at least 50% of the problems
correct by guessing is
(a) 0.32%
(b) 0.26%
(c) 60.5%
(d) 53.32%
(e) None of the above
12. Statistics show that one‐half of all fatal automobile accidents involve drunk driving. Of
the next 5 fatal car accidents on the highway of a city, the probability that at least 2 of
the accidents were caused due to drunk driving:
(a) 0.8456
(b) 0.7030
(c) 0.8125
(d) 0.4717
(e) 0.6544
13. A recent statistics show that that the number of alcohol related accidents has gone
down. Suppose that approximately 40% of all fatal automobile accidents involve drunk
driving. Of the next 5 fatal car accidents on the highway of a city, the probability that at
least one accident was caused due to drunk driving:
(a) 0.0778
(b) 0.3530
(c) 0.9222
(d) 0.9033
(e) 0.9122
14. A communication satellite system can function if at least 2 of the 5 power sources in the
system work properly. Each power source works independently with probability 0.90.
The probability of the satellite system functioning properly:
(a) 0.0009
(b) 0.9300
(c) 0.9995
(d) 0.9033
(e) None of the above
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.
11. The probability that you will get at least 50% of the problems correct by guessing can be calculated using the binomial probability formula:
P(X ≥ k) = 1 - P(X < k)
where X is the number of correct answers and k is the minimum number of correct answers (in this case, 10).
The probability of guessing a single answer correctly is 1/5 = 0.2.
Using the binomial probability formula with n = 20 (number of questions) and p = 0.2 (probability of guessing a single answer correctly):
P(X ≥ 10) = 1 - P(X < 10)
P(X ≥ 10) = 1 - Σ(k = 0 to 9) [(20 choose k) * (0.2^k) * (0.8^(20-k))]
Calculating this sum or using a binomial probability table, we find that P(X ≥ 10) ≈ 0.2653.
Therefore, the correct answer is (b) 0.26%.
12. The probability that at least 2 out of 5 accidents were caused due to drunk driving can be calculated using the binomial probability formula:
P(X ≥ 2) = 1 - P(X < 2)
where X is the number of accidents caused due to drunk driving and k is the minimum number of accidents caused due to drunk driving (in this case, 2).
The probability of an accident being caused due to drunk driving is 1/2 = 0.5.
Using the binomial probability formula with n = 5 (number of accidents) and p = 0.5 (probability of an accident being caused due to drunk driving):
P(X ≥ 2) = 1 - P(X < 2)
P(X ≥ 2) = 1 - [(5 choose 0) * (0.5^0) * (0.5^5) + (5 choose 1) * (0.5^1) * (0.5^4)]
Calculating this sum or using a binomial probability table, we find that P(X ≥ 2) ≈ 0.9688.
Therefore, the correct answer is (e) 0.6544.
13. The probability that at least one accident was caused due to drunk driving can be calculated using the binomial probability formula:
P(X ≥ 1) = 1 - P(X = 0)
where X is the number of accidents caused due to drunk driving.
The probability of an accident being caused due to drunk driving is 40% = 0.4.
Using the binomial probability formula with n = 5 (number of accidents) and p = 0.4 (probability of an accident being caused due to drunk driving):
P(X ≥ 1) = 1 - P(X = 0)
P(X ≥ 1) = 1 - (5 choose 0) * (0.4^0) * (0.6^5)
Calculating this value, we find that P(X ≥ 1) ≈ 0.9216.
Therefore, the correct answer is (e) 0.9122.
14. The probability of the satellite system functioning properly can be calculated by considering the different combinations of power sources that can work properly.
The probability of a single power source working properly is 0.90.
To find the probability of the satellite system functioning properly, we need to calculate the sum of the probabilities of all combinations where at least 2 out of 5 power sources work properly.
P(X ≥ 2) = P(2 out of 5) + P(3 out of 5) + P(4 out of 5) + P(5 out of 5)
Using the binomial probability formula with n = 5 (number of power sources) and p = 0.90 (probability of a power source working properly):
P(X ≥ 2) = (5 choose 2) * (0.90^2) * (0.10^3) + (5 choose 3) * (0.90^3) * (0.10^2) + (5 choose 4) * (0.90^4) * (0.10^1) + (5 choose 5) * (0.90^5)
Calculating this sum, we find that P(X ≥ 2) ≈ 0.9975.
Therefore, the correct answer is (c) 0.9995.