In a test, Chris has to answer two questions. The probability ... | Mathematics MCQ | Examoogle

Examoogle • User • info@examoogle.com • EE-2024-8821

Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked

MathematicsProbability

In a test, Chris has to answer two questions. The probability that he will answer the first question correctly is $\frac{1}{5}$ . If he answers the first question correctly, the probability that he will answer the second question correctly is $\frac{5}{6}$ . If Chris does not answer the first question correctly, the probability that he will answer the second question correctly is $\frac{1}{4}$ . Find the probability that Chris will answer the second question INCORRECTLY. [Note: Assume that each question will be answered— either correctly or incorrectly.]

$\frac{19}{30}$

$\frac{3}{5}$

$\frac{2}{3}$

$\frac{7}{10}$

Correct Answer

⚙️ TE • Technical Law of Total ProbabilityMathematicsProbability

Option A

$\frac{19}{30}$

Quick Summary:

Use the total probability tree: calculate the probability of the two successful paths to the second question's success, then subtract from 1 to get the failure probability.

📐MAMath SolutionLaw of Total Probability

📋 Given

P(Correct1) = 1/5, P(Correct2|Correct1) = 5/6, P(Correct2|Incorrect1) = 1/4

🔢 Formula Used

P(Ac)=1−P(A)P(A^c) = 1 - P(A)P(Ac)=1−P(A), P(C2)=P(C2∣C1)P(C1)+P(C2∣C1c)P(C1c)P(C_2) = P(C_2|C_1)P(C_1) + P(C_2|C_1^c)P(C_1^c)P(C2​)=P(C2​∣C1​)P(C1​)+P(C2​∣C1c​)P(C1c​)

⚡ Exam Hall Shortcut / Speed Trick

Use the total probability tree: calculate the probability of the two successful paths to the second question's success, then subtract from 1 to get the failure probability.

⚠️ Common Student Trap / Pitfall

Students often calculate the probability of answering the second question correctly but forget to subtract from 1 to find the 'incorrect' probability.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Calculate probabilities of first question outcomes

Probability of success $P(C_1) = 1/5$ . Thus, probability of failure $P(C_1^c) = 1 - 1/5 = 4/5$ .

P(C1c)=45P(C_1^c) = \frac{4}{5}P(C1c​)=54​

Calculate total probability of answering second question correctly

Using total probability: $P(C_2) = P(C_2|C_1)P(C_1) + P(C_2|C_1^c)P(C_1^c)$ . Substituting values: $(5/6 \times 1/5) + (1/4 \times 4/5)$ .

P(C2)=16+15=5+630=1130P(C_2) = \frac{1}{6} + \frac{1}{5} = \frac{5+6}{30} = \frac{11}{30}P(C2​)=61​+51​=305+6​=3011​

Calculate probability of answering second question incorrectly

The probability of failing the second question is $1 - P(C_2) = 1 - 11/30$ .

P(C2c)=1−1130=1930P(C_2^c) = 1 - \frac{11}{30} = \frac{19}{30}P(C2c​)=1−3011​=3019​

✅

A is correct because the total probability of answering the second question correctly is 11/30, making the probability of answering it incorrectly 1 - 11/30 = 19/30.

Core Concepts Used

Click any tag to open in AI Tutor

Total Probability Theorem Complementary Events Conditional Probability

💡 EXAM TIP

This concept is a subset of Bayes' Theorem; mastering the tree diagram method makes complex conditional probability problems much simpler to visualize.

Was this solution helpful?

Discussion (0)

Loading discussion...

Prev Next

Correct Answer

Related Questions

Discussion (0)