The odds against A solving a certain problem are 4 to 3 and the odds in favour of B solving the same problem are 7 to 5. The probability that the problem will be solved is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{16}{21} \)
Odds against A of 4 to 3 means 4 failures to 3 successes; odds in favour of B of 7 to 5 means 7 successes to 5 failures.
\( \displaystyle P(A)=\frac{3}{4+3}=\frac{3}{7} \)
\( \displaystyle P(B)=\frac{7}{7+5}=\frac{7}{12} \)
A and B work on the problem independently. The problem remains unsolved only if both fail:
\( \displaystyle P(\text{not solved})=P(A')\,P(B') \)
\( \displaystyle =\frac{4}{7}\times\frac{5}{12} \)
\( \displaystyle =\frac{20}{84}=\frac{5}{21} \)
\( \displaystyle P(\text{solved})=1-\frac{5}{21}=\frac{16}{21} \)
Why the other options are incorrect
Option (1) \( \displaystyle\frac{5}{21} \) — Incorrect. \( \frac{5}{21} \) is the probability that the problem is not solved (both fail).
Option (3) \( \displaystyle\frac{11}{21} \) — Incorrect. The probability is \( \frac{16}{21} \), not \( \frac{11}{21} \).
Option (4) \( \displaystyle\frac{13}{21} \) — Incorrect. The probability is \( \frac{16}{21} \), not \( \frac{13}{21} \).
Tip: Odds against a : b give probability \( \frac{b}{a+b} \); odds in favour a : b give \( \frac{a}{a+b} \).