Question:
Rooms in a hotel are numbered from 1 to 19. Rooms are allocated at random as guests arrive. The first guest to arrive is given a room which is a prime number. The probability that the second guest to arrive is given a room which is a prime number is
Rooms in a hotel are numbered from 1 to 19. Rooms are allocated at random as guests arrive. The first guest to arrive is given a room which is a prime number. The probability that the second guest to arrive is given a room which is a prime number is
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Be careful not to multiply the probabilities of the two events together! The question states that the first event has already happened ("The first guest to arrive is given a room which is a prime number"), so we do not factor in the probability of picking the first room. We only calculate the immediate probability for the second room from the updated pool.
Updated On: Jun 12, 2026
- $\frac{8}{19} \times \frac{7}{18}$
- $\frac{8}{19}$
- $\frac{8}{19} \times \frac{7}{19}$
- $\frac{7}{18}$
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
We are given a pool of hotel rooms labeled 1 through 19. The first guest is randomly assigned a room that is guaranteed to be a prime number. We want to find the probability that the second guest is also assigned a prime-numbered room.
Step 2: Key Formula or Approach:
Identify the total number of items available and the subset of favorable prime items. After the first independent allocation occurs, adjust both the total sample space and the favorable count downward to compute the classical probability: $$P = \frac{\text{Remaining Favorable Outcomes}}{\text{Remaining Total Outcomes}}$$
Step 3: Detailed Explanation:
1. List the prime numbers available in the set $\{1, 2, 3, \dots, 19\}$: $$\text{Primes} = \{2, 3, 5, 7, 11, 13, 17, 19\}$$ There are exactly 8 prime numbers out of 19 total rooms. 2. The first guest arrives and is allocated a room that is a prime number.
This reduces the total number of available rooms from 19 down to 18 ($19 - 1 = 18$). This also reduces the number of remaining prime rooms from 8 down to 7 ($8 - 1 = 7$). 3. Calculate the conditional probability for the second guest's room selection: $$P(\text{Second is Prime} \mid \text{First is Prime}) = \frac{7}{18}$$
Step 4: Final Answer:
The probability that the second guest receives a prime room is $\frac{7}{18}$, which corresponds to option (D).
We are given a pool of hotel rooms labeled 1 through 19. The first guest is randomly assigned a room that is guaranteed to be a prime number. We want to find the probability that the second guest is also assigned a prime-numbered room.
Step 2: Key Formula or Approach:
Identify the total number of items available and the subset of favorable prime items. After the first independent allocation occurs, adjust both the total sample space and the favorable count downward to compute the classical probability: $$P = \frac{\text{Remaining Favorable Outcomes}}{\text{Remaining Total Outcomes}}$$
Step 3: Detailed Explanation:
1. List the prime numbers available in the set $\{1, 2, 3, \dots, 19\}$: $$\text{Primes} = \{2, 3, 5, 7, 11, 13, 17, 19\}$$ There are exactly 8 prime numbers out of 19 total rooms. 2. The first guest arrives and is allocated a room that is a prime number.
This reduces the total number of available rooms from 19 down to 18 ($19 - 1 = 18$). This also reduces the number of remaining prime rooms from 8 down to 7 ($8 - 1 = 7$). 3. Calculate the conditional probability for the second guest's room selection: $$P(\text{Second is Prime} \mid \text{First is Prime}) = \frac{7}{18}$$
Step 4: Final Answer:
The probability that the second guest receives a prime room is $\frac{7}{18}$, which corresponds to option (D).
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