Question
The probability that a candidate passes an exam is 1/5 find the probability that if
Two candidates are selected at random, only one passes the examination.
Three candidates selected at random all pass the examination
Two candidates are selected at random, only one passes the examination.
Three candidates selected at random all pass the examination
Answers
GPT-4o mini
To solve the problem, we need to use the concepts of probability and the binomial distribution.
**1. Probability that only one of the two candidates passes the exam:**
Let \( p = \frac{1}{5} \) be the probability that a candidate passes the exam and \( q = 1 - p = \frac{4}{5} \) be the probability that a candidate fails.
We want to find the probability that exactly one of the two candidates passes. This scenario can happen in two ways:
- The first candidate passes and the second fails.
- The first candidate fails and the second passes.
The formula for the probability of exactly one candidate passing (out of two candidates) is given by:
\[
P(\text{exactly one passes}) = P(\text{one passes, one fails}) + P(\text{one fails, one passes})
\]
Calculating this:
\[
P(\text{one passes, one fails}) = p \cdot q = \left( \frac{1}{5} \right) \cdot \left( \frac{4}{5} \right) = \frac{4}{25}
\]
\[
P(\text{one fails, one passes}) = q \cdot p = \left( \frac{4}{5} \right) \cdot \left( \frac{1}{5} \right) = \frac{4}{25}
\]
So, the total probability that exactly one of the two candidates passes is:
\[
P(\text{exactly one passes}) = \frac{4}{25} + \frac{4}{25} = \frac{8}{25}
\]
**2. Probability that all three candidates pass the examination:**
To find the probability that all three candidates pass, we calculate:
\[
P(\text{all three pass}) = p^3 = \left( \frac{1}{5} \right)^3 = \frac{1}{125}
\]
**Summary of results:**
- The probability that only one of the two candidates passes the examination is \( \frac{8}{25} \).
- The probability that all three candidates pass the examination is \( \frac{1}{125} \).
**1. Probability that only one of the two candidates passes the exam:**
Let \( p = \frac{1}{5} \) be the probability that a candidate passes the exam and \( q = 1 - p = \frac{4}{5} \) be the probability that a candidate fails.
We want to find the probability that exactly one of the two candidates passes. This scenario can happen in two ways:
- The first candidate passes and the second fails.
- The first candidate fails and the second passes.
The formula for the probability of exactly one candidate passing (out of two candidates) is given by:
\[
P(\text{exactly one passes}) = P(\text{one passes, one fails}) + P(\text{one fails, one passes})
\]
Calculating this:
\[
P(\text{one passes, one fails}) = p \cdot q = \left( \frac{1}{5} \right) \cdot \left( \frac{4}{5} \right) = \frac{4}{25}
\]
\[
P(\text{one fails, one passes}) = q \cdot p = \left( \frac{4}{5} \right) \cdot \left( \frac{1}{5} \right) = \frac{4}{25}
\]
So, the total probability that exactly one of the two candidates passes is:
\[
P(\text{exactly one passes}) = \frac{4}{25} + \frac{4}{25} = \frac{8}{25}
\]
**2. Probability that all three candidates pass the examination:**
To find the probability that all three candidates pass, we calculate:
\[
P(\text{all three pass}) = p^3 = \left( \frac{1}{5} \right)^3 = \frac{1}{125}
\]
**Summary of results:**
- The probability that only one of the two candidates passes the examination is \( \frac{8}{25} \).
- The probability that all three candidates pass the examination is \( \frac{1}{125} \).