Pre-med style conceptual questions on probability fundamentals (complement, inclusion-exclusion, independence, conditional probability), counting (permutations, combinations), Bayes' theorem and base rates, descriptive statistics (mean, median, mode, standard deviation), normal distributions, hypothesis testing, correlation vs causation, and experimental design.
The probability of an event occurring is 0.3. What is the probability that the event does NOT occur?
Source: entermedschool.org
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src="https://entermedschool.org/embed/questions#{"title":"Statistics and Probability","questions":[{"prompt":"The probability of an event occurring is 0.3. What is the probability that the event does NOT occur?","options":[{"label":"A","body":"0.3","isCorrect":false},{"label":"B","body":"0.7","isCorrect":true},{"label":"C","body":"0.03","isCorrect":false},{"label":"D","body":"1.3","isCorrect":false},{"label":"E","body":"Cannot be determined without more information","isCorrect":false}],"explanation":"By the complement rule, P(not A) = 1 - P(A) = 1 - 0.3 = 0.7. The sum of an event and its complement must equal 1."},{"prompt":"In a class, 60% of students play football, 40% play basketball, and 25% play both. What percentage of students play at least one of these sports?","options":[{"label":"A","body":"100%","isCorrect":false},{"label":"B","body":"75%","isCorrect":true},{"label":"C","body":"85%","isCorrect":false},{"label":"D","body":"65%","isCorrect":false},{"label":"E","body":"50%","isCorrect":false}],"explanation":"Using inclusion-exclusion: P(F or B) = P(F) + P(B) - P(F and B) = 60% + 40% - 25% = 75%. We subtract the intersection to avoid double-counting."},{"prompt":"A bag contains 5 red and 3 blue marbles. If two marbles are drawn WITHOUT replacement, the probability of drawing two red marbles is:","options":[{"label":"A","body":"25/64","isCorrect":false},{"label":"B","body":"5/14","isCorrect":true},{"label":"C","body":"5/8 × 5/8","isCorrect":false},{"label":"D","body":"10/56","isCorrect":false},{"label":"E","body":"1/2","isCorrect":false}],"explanation":"Without replacement: P(both red) = (5/8) × (4/7) = 20/56 = 5/14. After removing one red, there are 4 red left out of 7 total. Option A uses replacement (5/8 × 5/8); D is 10/56 which equals 5/28, not 5/14."},{"prompt":"Events A and B are independent. If P(A) = 0.4 and P(B) = 0.5, what is P(A and B)?","options":[{"label":"A","body":"0.9","isCorrect":false},{"label":"B","body":"0.1","isCorrect":false},{"label":"C","body":"0.2","isCorrect":true},{"label":"D","body":"0.45","isCorrect":false},{"label":"E","body":"Cannot be calculated without knowing P(A|B)","isCorrect":false}],"explanation":"For independent events, P(A and B) = P(A) × P(B) = 0.4 × 0.5 = 0.2. Independence means knowing B occurred doesn't change the probability of A."},{"prompt":"If P(A) = 0.6, P(B) = 0.5, and P(A and B) = 0.3, what is P(A|B)?","options":[{"label":"A","body":"0.3","isCorrect":false},{"label":"B","body":"0.5","isCorrect":false},{"label":"C","body":"0.6","isCorrect":true},{"label":"D","body":"0.8","isCorrect":false},{"label":"E","body":"1.1","isCorrect":false}],"explanation":"Conditional probability: P(A|B) = P(A and B)/P(B) = 0.3/0.5 = 0.6. This tells us the probability of A given that B has occurred."},{"prompt":"Two events are mutually exclusive. Which statement must be true?","options":[{"label":"A","body":"P(A and B) = P(A) × P(B)","isCorrect":false},{"label":"B","body":"P(A or B) = P(A) + P(B)","isCorrect":true},{"label":"C","body":"P(A|B) = P(A)","isCorrect":false},{"label":"D","body":"If one occurs, the other must also occur","isCorrect":false},{"label":"E","body":"They must have the same probability","isCorrect":false}],"explanation":"Mutually exclusive means P(A and B) = 0 (they cannot both occur). Therefore P(A or B) = P(A) + P(B) - 0 = P(A) + P(B). Option A describes independence, not mutual exclusivity."},{"prompt":"A fair coin is flipped 5 times and lands heads each time. What is the probability that the 6th flip will be heads?","options":[{"label":"A","body":"Less than 1/2 because tails is 'due'","isCorrect":false},{"label":"B","body":"More than 1/2 because the coin is 'hot'","isCorrect":false},{"label":"C","body":"Exactly 1/2","isCorrect":true},{"label":"D","body":"1/64","isCorrect":false},{"label":"E","body":"5/6","isCorrect":false}],"explanation":"Each flip is independent. Previous outcomes don't affect future ones. The probability remains 1/2. Thinking otherwise is the gambler's fallacy. Option D (1/64) would be the probability of 6 heads in a row before flipping."},{"prompt":"A jar contains 4 red, 3 blue, and 2 green marbles. If one marble is drawn at random, what is P(red OR green)?","options":[{"label":"A","body":"4/9","isCorrect":false},{"label":"B","body":"2/9","isCorrect":false},{"label":"C","body":"6/9","isCorrect":true},{"label":"D","body":"8/81","isCorrect":false},{"label":"E","body":"1/9","isCorrect":false}],"explanation":"Red and green are mutually exclusive (can't draw both at once), so P(red or green) = P(red) + P(green) = 4/9 + 2/9 = 6/9 = 2/3. Option D incorrectly multiplies the probabilities."},{"prompt":"In a standard deck of 52 cards, what is the probability of drawing a King OR a Heart?","options":[{"label":"A","body":"17/52","isCorrect":false},{"label":"B","body":"16/52","isCorrect":true},{"label":"C","body":"4/52","isCorrect":false},{"label":"D","body":"13/52","isCorrect":false},{"label":"E","body":"1/52","isCorrect":false}],"explanation":"Using inclusion-exclusion: P(King or Heart) = P(King) + P(Heart) - P(King and Heart) = 4/52 + 13/52 - 1/52 = 16/52. We subtract the King of Hearts to avoid counting it twice."},{"prompt":"A die is rolled twice. What is the probability of getting a sum of 7?","options":[{"label":"A","body":"1/36","isCorrect":false},{"label":"B","body":"6/36","isCorrect":true},{"label":"C","body":"7/36","isCorrect":false},{"label":"D","body":"1/6","isCorrect":false},{"label":"E","body":"7/12","isCorrect":false}],"explanation":"There are 36 equally likely outcomes. Pairs summing to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes. P = 6/36 = 1/6. Note that D is also 1/6, which is correct - B and D are equivalent."},{"prompt":"Three cards are drawn from a deck without replacement. Compared to drawing with replacement, the probability that all three are aces is:","options":[{"label":"A","body":"Higher with replacement","isCorrect":true},{"label":"B","body":"Higher without replacement","isCorrect":false},{"label":"C","body":"The same either way","isCorrect":false},{"label":"D","body":"Cannot be compared","isCorrect":false},{"label":"E","body":"Zero in both cases","isCorrect":false}],"explanation":"With replacement: (4/52)^3 = 64/140608. Without replacement: (4/52)(3/51)(2/50) = 24/132600. The with-replacement probability is higher because removing aces reduces future chances."},{"prompt":"If events A and B are independent, which must be true?","options":[{"label":"A","body":"P(A and B) = 0","isCorrect":false},{"label":"B","body":"P(A or B) = P(A) + P(B)","isCorrect":false},{"label":"C","body":"P(A|B) = P(A)","isCorrect":true},{"label":"D","body":"A and B cannot both occur","isCorrect":false},{"label":"E","body":"P(A) = P(B)","isCorrect":false}],"explanation":"Independence means knowing B occurred doesn't change the probability of A: P(A|B) = P(A). Options A and D describe mutually exclusive events. Option B is true only for mutually exclusive events."},{"prompt":"A test has 5 true/false questions. If a student guesses randomly on all questions, what is the probability of getting all 5 correct?","options":[{"label":"A","body":"1/5","isCorrect":false},{"label":"B","body":"1/10","isCorrect":false},{"label":"C","body":"1/25","isCorrect":false},{"label":"D","body":"1/32","isCorrect":true},{"label":"E","body":"1/2","isCorrect":false}],"explanation":"Each question has P = 1/2 of being correct. For all 5: (1/2)^5 = 1/32. This assumes independent guessing."},{"prompt":"A family has 3 children. Assuming equal probability of boy or girl for each child, what is the probability that all three are the same gender?","options":[{"label":"A","body":"1/8","isCorrect":false},{"label":"B","body":"1/4","isCorrect":true},{"label":"C","body":"1/3","isCorrect":false},{"label":"D","body":"1/2","isCorrect":false},{"label":"E","body":"3/8","isCorrect":false}],"explanation":"P(all boys) = (1/2)^3 = 1/8. P(all girls) = 1/8. P(all same) = 1/8 + 1/8 = 2/8 = 1/4. These are mutually exclusive outcomes."},{"prompt":"Events A and B are such that P(A) = 0.4, P(B) = 0.3, and P(A and B) = 0.12. Are A and B independent?","options":[{"label":"A","body":"Yes, because P(A and B) = P(A) × P(B)","isCorrect":true},{"label":"B","body":"No, because P(A) ≠ P(B)","isCorrect":false},{"label":"C","body":"No, because P(A and B) ≠ 0","isCorrect":false},{"label":"D","body":"Yes, because P(A or B) = P(A) + P(B)","isCorrect":false},{"label":"E","body":"Cannot be determined","isCorrect":false}],"explanation":"Events are independent if P(A and B) = P(A) × P(B). Here: 0.4 × 0.3 = 0.12 = P(A and B). So yes, they are independent. B confuses independence with equal probability; C confuses with mutual exclusivity."},{"prompt":"In how many ways can 5 different books be arranged on a shelf?","options":[{"label":"A","body":"5","isCorrect":false},{"label":"B","body":"25","isCorrect":false},{"label":"C","body":"120","isCorrect":true},{"label":"D","body":"10","isCorrect":false},{"label":"E","body":"32","isCorrect":false}],"explanation":"This is a permutation of 5 distinct objects: 5! = 5 × 4 × 3 × 2 × 1 = 120. Order matters when arranging on a shelf."},{"prompt":"A committee of 3 people must be chosen from a group of 7. How many different committees are possible?","options":[{"label":"A","body":"21","isCorrect":false},{"label":"B","body":"35","isCorrect":true},{"label":"C","body":"210","isCorrect":false},{"label":"D","body":"343","isCorrect":false},{"label":"E","body":"7","isCorrect":false}],"explanation":"Order doesn't matter for a committee, so we use combinations: C(7,3) = 7!/(3!×4!) = (7×6×5)/(3×2×1) = 35. Option C (210) is the permutation P(7,3)."},{"prompt":"How many 4-digit PINs are possible if repetition of digits is allowed?","options":[{"label":"A","body":"40","isCorrect":false},{"label":"B","body":"5,040","isCorrect":false},{"label":"C","body":"10,000","isCorrect":true},{"label":"D","body":"24","isCorrect":false},{"label":"E","body":"1,000","isCorrect":false}],"explanation":"Each of 4 positions can be any of 10 digits (0-9): 10^4 = 10,000. Option B (5,040) would be without repetition: 10×9×8×7."},{"prompt":"A president, vice president, and treasurer must be elected from 8 candidates. How many ways can these positions be filled?","options":[{"label":"A","body":"24","isCorrect":false},{"label":"B","body":"56","isCorrect":false},{"label":"C","body":"336","isCorrect":true},{"label":"D","body":"512","isCorrect":false},{"label":"E","body":"8","isCorrect":false}],"explanation":"Order matters (different positions), so P(8,3) = 8×7×6 = 336. Option B (56) is C(8,3), which ignores that positions are distinct."},{"prompt":"How many ways can the letters in 'BOOK' be arranged?","options":[{"label":"A","body":"24","isCorrect":false},{"label":"B","body":"12","isCorrect":true},{"label":"C","body":"4","isCorrect":false},{"label":"D","body":"6","isCorrect":false},{"label":"E","body":"16","isCorrect":false}],"explanation":"BOOK has 4 letters with O repeated twice. The formula is 4!/2! = 24/2 = 12. We divide by 2! because swapping the two O's doesn't create a new arrangement."}]}"
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