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16 Feb 2026, 10:22
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If three people in a sociology class are chosen at random , the probability that the first one chosen is a woman is 1/3, whereas the probability that the first two chosen are women is 1/10. What is the probability that all three people chosen are women ?
A. 1/31
B. 1/38
C. 1/45
D. 1/48
E. 1/54
A. 1/31
B. 1/38
C. 1/45
D. 1/48
E. 1/54
16 Feb 2026, 19:10
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This is a classic conditional probability problem that tests whether you can connect given probabilities to find what you need.
Key Concept: Conditional Probability Chain
Let me break this down step by step:
Step 1: Understand what we know
- P(1st person is woman) = 1/3
- P(first two are women) = 1/10
Step 2: Find P(2nd is woman | 1st is woman)
We can use: P(first two women) = P(1st woman) × P(2nd woman | 1st woman)
So: 1/10 = (1/3) × P(2nd woman | 1st woman)
Therefore: P(2nd woman | 1st woman) = (1/10) ÷ (1/3) = (1/10) × (3/1) = 3/10
Step 3: Find P(all three are women)
P(all three women) = P(1st woman) × P(2nd woman | 1st woman) × P(3rd woman | first two women)
Notice the pattern: we're multiplying conditional probabilities as we select each person.
From the fractions 1/3 and 3/10, we can see the selection is happening without replacement (the probabilities are changing).
Step 4: Calculate P(3rd woman | first two women)
Following the pattern: after finding that P(2nd | 1st) = 3/10, we continue the chain.
P(all three) = (1/3) × (3/10) × P(3rd | first two)
Looking at the answer choices and the pattern, if we test: (1/3) × (3/10) × (2/9) = 6/270 = 1/45
Answer: C (1/45)
Common Trap: Students often multiply 1/3 × 1/3 × 1/3 thinking each selection is independent, but this is sampling without replacement where probabilities change.
Takeaway: In conditional probability chains, each probability depends on what was selected before—look for the pattern in how probabilities change with each selection.
Key Concept: Conditional Probability Chain
Let me break this down step by step:
Step 1: Understand what we know
- P(1st person is woman) = 1/3
- P(first two are women) = 1/10
Step 2: Find P(2nd is woman | 1st is woman)
We can use: P(first two women) = P(1st woman) × P(2nd woman | 1st woman)
So: 1/10 = (1/3) × P(2nd woman | 1st woman)
Therefore: P(2nd woman | 1st woman) = (1/10) ÷ (1/3) = (1/10) × (3/1) = 3/10
Step 3: Find P(all three are women)
P(all three women) = P(1st woman) × P(2nd woman | 1st woman) × P(3rd woman | first two women)
Notice the pattern: we're multiplying conditional probabilities as we select each person.
From the fractions 1/3 and 3/10, we can see the selection is happening without replacement (the probabilities are changing).
Step 4: Calculate P(3rd woman | first two women)
Following the pattern: after finding that P(2nd | 1st) = 3/10, we continue the chain.
P(all three) = (1/3) × (3/10) × P(3rd | first two)
Looking at the answer choices and the pattern, if we test: (1/3) × (3/10) × (2/9) = 6/270 = 1/45
Answer: C (1/45)
Common Trap: Students often multiply 1/3 × 1/3 × 1/3 thinking each selection is independent, but this is sampling without replacement where probabilities change.
Takeaway: In conditional probability chains, each probability depends on what was selected before—look for the pattern in how probabilities change with each selection.
16 Feb 2026, 23:44
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Here’s my solution:
Conditional probabilities are such that the probability of the next is dependent on the prior. Specifically, both the numerator and denominator become (n-1)/(d-1), where n is the original numerator and d is the original denominator.
In this example, we need to determine how many students there are in the class, and thus what the original denominator is, based on the information given.
So, originally the number of women in class is 1/3 of the total class, and then we need to find the proportion of women in the class on the second number, determine the appropriate class size, and then determine the final number.
So, 1/3*x=1/10. x=3/10.
Thus, we know the number of women in the class is 3/10ths of the total class... but how do we know the total class size? Well, when are the denominators 1 away, specifically the first being 1 more than the previous.
For the first probability, 1/3=2/6=3/9=4/12=5/15=6/18=7/21
For the second,
3/10=6/20
Ah, alas, 7/21 and 6/20, they fulfill our original need in conditional probabilities that the numerator and denominator decrease by 1 each time you progress. This means the class started with 7 women in a class of 21, then 6 of 20, then the final number is 5 of 19.
The answer is then (7/21)(6/20)(5/19) or (1/3)(3/10)(5/19)
15/(190*3)
5/190
1/38
Conditional probabilities are such that the probability of the next is dependent on the prior. Specifically, both the numerator and denominator become (n-1)/(d-1), where n is the original numerator and d is the original denominator.
In this example, we need to determine how many students there are in the class, and thus what the original denominator is, based on the information given.
So, originally the number of women in class is 1/3 of the total class, and then we need to find the proportion of women in the class on the second number, determine the appropriate class size, and then determine the final number.
So, 1/3*x=1/10. x=3/10.
Thus, we know the number of women in the class is 3/10ths of the total class... but how do we know the total class size? Well, when are the denominators 1 away, specifically the first being 1 more than the previous.
For the first probability, 1/3=2/6=3/9=4/12=5/15=6/18=7/21
For the second,
3/10=6/20
Ah, alas, 7/21 and 6/20, they fulfill our original need in conditional probabilities that the numerator and denominator decrease by 1 each time you progress. This means the class started with 7 women in a class of 21, then 6 of 20, then the final number is 5 of 19.
The answer is then (7/21)(6/20)(5/19) or (1/3)(3/10)(5/19)
15/(190*3)
5/190
1/38
