The odds against an event is 7: 4 and the odds in favour of an... | Mathematics MCQ | Examoogle

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The odds against an event is 7: 4 and the odds in favour of another event is 8: 5. If both the events are independent, then the probability that at least one of the event will happen is:

$\frac{88}{143}$

$\frac{76}{143}$

$\frac{35}{143}$

$\frac{108}{143}$

Correct Answer

⚙️ TE • Technical Complementary ProbabilityMathematicsProbability

Option D

$\frac{108}{143}$

Quick Summary:

Calculate the probability of non-occurrence for both events ( $P(A^c) = \frac{7}{7+4} = \frac{7}{11}$ and $P(B^c) = \frac{5}{8+5} = \frac{5}{13}$ ) and subtract their product from 1.

📐MAMath SolutionComplementary Probability

📋 Given

Odds against event A = 7:4, odds in favour of event B = 8:5.

🔢 Formula Used

P(A∪B)=1−P(Ac)×P(Bc)P(A \cup B) = 1 - P(A^c) \times P(B^c)P(A∪B)=1−P(Ac)×P(Bc)

⚡ Exam Hall Shortcut / Speed Trick

Calculate the probability of non-occurrence for both events ( $P(A^c) = \frac{7}{7+4} = \frac{7}{11}$ and $P(B^c) = \frac{5}{8+5} = \frac{5}{13}$ ) and subtract their product from 1.

⚠️ Common Student Trap / Pitfall

Confusing 'odds against' with 'odds in favour' and forgetting to calculate the complement ( $P(A^c)$ instead of $P(A)$ ).

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Calculate probabilities of non-occurrence

For event A, odds against are 7:4, so $P(A^c) = \frac{7}{7+4} = \frac{7}{11}$ . For event B, odds in favour are 8:5, so the probability of event B is $\frac{8}{13}$ , meaning $P(B^c) = 1 - \frac{8}{13} = \frac{5}{13}$ .

P(Ac)=711,P(Bc)=513P(A^c) = \frac{7}{11}, P(B^c) = \frac{5}{13}P(Ac)=117​,P(Bc)=135​

Find probability that neither event occurs

Since the events are independent, the probability that neither occurs is the product of their individual complement probabilities.

P(Ac∩Bc)=711×513=35143P(A^c \cap B^c) = \frac{7}{11} \times \frac{5}{13} = \frac{35}{143}P(Ac∩Bc)=117​×135​=14335​

Calculate probability of at least one event

The probability that at least one event occurs is the complement of the probability that neither event occurs.

P(A∪B)=1−35143=143−35143=108143P(A \cup B) = 1 - \frac{35}{143} = \frac{143-35}{143} = \frac{108}{143}P(A∪B)=1−14335​=143143−35​=143108​

✅

D is correct because the calculated probability of at least one event occurring is $\frac{108}{143}$ .

Core Concepts Used

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Independent Events Odds vs Probability Complementary Events

💡 EXAM TIP

This concept of '1 - P(none)' is a standard technique in binomial distribution problems and reliability engineering to quickly find the probability of 'at least one success'.

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