7. An AQR instructor develops an exam with 10 questions. Each question has 4 possible answers, but only one is comect. A student who didn't study randomly guesses on each question Answer the following questions. a) What is the probability that the student will guess exactly 5 correctly? b) What is the probability that the student will guess exactly 7 correctly? c) What is the probability that the student will score less than 7 correct? d) What is the probability that the student will score 7 or above? c) What is the probability of getting 100% f) If there are five questions, what is the probability of getting 100% 8. Kelly is an 85% foul shooter, a) what is the probability that she makes exactly 7 out of 10 attempts from the foul line? b) What is the probability that she makes 5 or less foul shots out of 10 attempts? c) What is the probability that she makes 8 or more out of 10 attempts? 9. Approximately 4% of the eggs in a store are cracked. If you buy two dozen eggs,what is the probability that a) none of your eggs are cracked. b) at least one of your eggs is cracked. c) exactly two of your eggs are cracked. 10. A baseball player has a batting average of 0.285. Find each of the following probabilities a) Exactly 4 hits in her next 10 times at bat. b) More than 4 hits in her next 10 times at bat. c) Less than 4 hits in her next 10 times at bat.

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7. An AQR instructor develops an exam with 10 questions. Each question has 4 possible answers, but only one is comect. A student who didn't study randomly guesses on each question Answer the following questions. a) What is the probability that the student will guess exactly 5 correctly? b) What is the probability that the student will guess exactly 7 correctly? c) What is the probability that the student will score less than 7 correct? d) What is the probability that the student will score 7 or above? c) What is the probability of getting 100% f) If there are five questions, what is the probability of getting 100% 8. Kelly is an 85% foul shooter, a) what is the probability that she makes exactly 7 out of 10 attempts from the foul line? b) What is the probability that she makes 5 or less foul shots out of 10 attempts? c) What is the probability that she makes 8 or more out of 10 attempts? 9. Approximately 4% of the eggs in a store are cracked. If you buy two dozen eggs,what is the probability that a) none of your eggs are cracked. b) at least one of your eggs is cracked. c) exactly two of your eggs are cracked. 10. A baseball player has a batting average of 0.285. Find each of the following probabilities a) Exactly 4 hits in her next 10 times at bat. b) More than 4 hits in her next 10 times at bat. c) Less than 4 hits in her next 10 times at bat.

Answer 1

### a) 0.0584 ### b) 0.0031 ### c) 0.9965 ### d) 0.0035 ### e) 0.000001 ### f) 0.001 ### 8a) 0.123 ### 8b) 0.008 ### 8c) 0.82 ### 9a) 0.37 ### 9b) 0.63 ### 9c) 0.18 ### 10a) 0.20 ### 10b) 0.16 ### 10c) 0.64

Answer 2

## Step 1: Probability of guessing exactly k questions correctly ### We use the binomial probability formula: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$, where $n$ is the number of trials, $k$ is the number of successes, and $p$ is the probability of success on a single trial. In this case, $n=10$ and $p = \frac{1}{4} = 0.25$. ## Step 2: Calculate a) P(X=5) ### $P(X=5) = \binom{10}{5} (0.25)^5 (0.75)^5 \approx 0.0584$ ## Step 3: Calculate b) P(X=7) ### $P(X=7) = \binom{10}{7} (0.25)^7 (0.75)^3 \approx 0.0031$ ## Step 4: Calculate c) P(X<7) ### $P(X<7) = 1 - P(X \ge 7) = 1 - [P(X=7) + P(X=8) + P(X=9) + P(X=10)]$. Calculating the individual probabilities and summing them, we get $P(X<7) \approx 1 - (0.0031 + 0.0004 + 0.0000 + 0.0000) \approx 0.9965$. ## Step 5: Calculate d) P(X≥7) ### $P(X \ge 7) = P(X=7) + P(X=8) + P(X=9) + P(X=10) \approx 0.0031 + 0.0004 + 0.0000 + 0.0000 \approx 0.0035$. ## Step 6: Calculate e) P(X=10) ### $P(X=10) = \binom{10}{10} (0.25)^{10} (0.75)^0 \approx 9.54 \times 10^{-7} \approx 0.000001$ ## Step 7: Calculate f) P(X=5) with n=5 ### With $n=5$, $P(X=5) = \binom{5}{5} (0.25)^5 (0.75)^0 \approx 0.00098 \approx 0.001$ ## Step 8: Kelly's foul shots ### We use the binomial probability formula again, with $n=10$ and $p=0.85$. ## Step 9: Calculate 8a) P(X=7) ### $P(X=7) = \binom{10}{7} (0.85)^7 (0.15)^3 \approx 0.123$ ## Step 10: Calculate 8b) P(X≤5) ### $P(X \le 5) = \sum_{k=0}^{5} \binom{10}{k} (0.85)^k (0.15)^{10-k} \approx 0.008$ ## Step 11: Calculate 8c) P(X≥8) ### $P(X \ge 8) = \sum_{k=8}^{10} \binom{10}{k} (0.85)^k (0.15)^{10-k} \approx 0.82$ ## Step 12: Cracked eggs ### We use the binomial probability formula with $n=24$ (two dozen) and $p=0.04$. ## Step 13: Calculate 9a) P(X=0) ### $P(X=0) = \binom{24}{0} (0.04)^0 (0.96)^{24} \approx 0.37$ ## Step 14: Calculate 9b) P(X≥1) ### $P(X \ge 1) = 1 - P(X=0) = 1 - 0.37 \approx 0.63$ ## Step 15: Calculate 9c) P(X=2) ### $P(X=2) = \binom{24}{2} (0.04)^2 (0.96)^{22} \approx 0.18$ ## Step 16: Baseball player's hits ### We use the binomial probability formula with $n=10$ and $p=0.285$. ## Step 17: Calculate 10a) P(X=4) ### $P(X=4) = \binom{10}{4} (0.285)^4 (0.715)^6 \approx 0.20$ ## Step 18: Calculate 10b) P(X>4) ### $P(X>4) = \sum_{k=5}^{10} \binom{10}{k} (0.285)^k (0.715)^{10-k} \approx 0.16$ ## Step 19: Calculate 10c) P(X<4) ### $P(X<4) = \sum_{k=0}^{3} \binom{10}{k} (0.285)^k (0.715)^{10-k} \approx 0.64$

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