Question:
Person A can solve \(90\%\) of the problems given in the book and Person B can solve \(70\%\). Then the probability that at least one of them will solve the problem selected at random from the book is:
Person A can solve \(90\%\) of the problems given in the book and Person B can solve \(70\%\). Then the probability that at least one of them will solve the problem selected at random from the book is:
Show Hint
For probability questions involving "at least one", it is usually easier to use the complement rule:
\[
P(\text{at least one})
=
1-P(\text{none})
\]
This method avoids lengthy calculations.
Updated On: Jun 26, 2026
- \(0.16\)
- \(0.69\)
- \(0.97\)
- \(0.20\)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Define the given probabilities.
Let \[ P(A)=0.90 \] be the probability that Person A solves a randomly selected problem.
Similarly, \[ P(B)=0.70 \] is the probability that Person B solves the same problem.
We are required to find the probability that at least one of them solves the problem.
Step 2: Use the complementary event.
The event "at least one solves the problem" is the complement of the event "neither A nor B solves the problem".
Therefore, \[ P(\text{at least one solves}) = 1-P(\text{neither solves}) \]
Step 3: Find the probability that neither solves the problem.
The probability that A does not solve the problem is \[ 1-0.90=0.10 \] The probability that B does not solve the problem is \[ 1-0.70=0.30 \] Assuming the solving abilities are independent, \[ P(\text{neither solves}) = 0.10\times 0.30 \] \[ = 0.03 \]
Step 4: Compute the required probability.
Hence, \[ P(\text{at least one solves}) = 1-0.03 \] \[ = 0.97 \]
Step 5: Final conclusion.
Therefore, the required probability is \[ \boxed{0.97} \] Hence, the correct option is \[ \boxed{(3)\ 0.97} \]
Let \[ P(A)=0.90 \] be the probability that Person A solves a randomly selected problem.
Similarly, \[ P(B)=0.70 \] is the probability that Person B solves the same problem.
We are required to find the probability that at least one of them solves the problem.
Step 2: Use the complementary event.
The event "at least one solves the problem" is the complement of the event "neither A nor B solves the problem".
Therefore, \[ P(\text{at least one solves}) = 1-P(\text{neither solves}) \]
Step 3: Find the probability that neither solves the problem.
The probability that A does not solve the problem is \[ 1-0.90=0.10 \] The probability that B does not solve the problem is \[ 1-0.70=0.30 \] Assuming the solving abilities are independent, \[ P(\text{neither solves}) = 0.10\times 0.30 \] \[ = 0.03 \]
Step 4: Compute the required probability.
Hence, \[ P(\text{at least one solves}) = 1-0.03 \] \[ = 0.97 \]
Step 5: Final conclusion.
Therefore, the required probability is \[ \boxed{0.97} \] Hence, the correct option is \[ \boxed{(3)\ 0.97} \]
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