Conditional Probability

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Conditional Probability · Grade 10 · Worksheet 3

1. A circle is inscribed in a right triangle with legs measuring 6 cm and 8 cm. The circle touches all three sides of the triangle. What is the radius of the inscribed circle? Answer: 2 Solution: We have a right triangle with legs 6 cm and 8 cm. A circle is inscribed inside it, touching all three sides. This is called the incircle of the triangle.
   Full step-by-step solution

   Step 1: Understand the problem We have a right triangle with legs 6 cm and 8 cm. A circle is inscribed inside it, touching all three sides. This is called the incircle of the triangle. The radius of the incircle is what we need to find. Step 2: Recall the formula for the inradius of a right triangle For a right triangle with legs a and b, and hypotenuse c, the inradius r is given by: r = (a + b - c) / 2 This formula comes from the fact that the area of the triangle equals r times the semiperimeter: Area = r * s, where s = (a + b + c)/2. Step 3: Find the hypotenuse Using the Pythagorean theorem: c^2 = a^2 + b^2 c^2 = 6^2 + 8^2 = 36 + 64 = 100 c = sqrt(100) = 10 cm Step 4: Apply the inradius formula r = (a + b - c) / 2 r = (6 + 8 - 10) / 2 r = (4) / 2 r = 2 cm Step 5: Verify with the area method Area of triangle = (1/2) * base * height = (1/2) * 6 * 8 = 24 cm^2 Semiperimeter s = (6 + 8 + 10) / 2 = 24 / 2 = 12 cm Area = r * s 24 = r * 12 r = 24 / 12 = 2 cm Both methods give the same answer. Final answer: The radius of the inscribed circle is 2 cm.

2. Matiu runs a community garden. He surveyed 240 members to find out their gardening preferences. He found that 144 members grow vegetables, 108 members grow flowers, and 72 members grow both vegetables and flowers. If a randomly selected member from the survey grows vegetables, what is the probability that they also grow flowers? Express your answer as a simplified fraction. Answer: 1/2 Solution: Define events. Let V be the event that a member grows vegetables, and F be the event that a member grows flowers. Use the conditional probability formula: P(F|V) = P(F and V) / P(V).
   Full step-by-step solution

   Step 1: Define events. Let V be the event that a member grows vegetables, and F be the event that a member grows flowers. We want P(F|V). Step 2: Use the conditional probability formula: P(F|V) = P(F and V) / P(V). Step 3: Find P(F and V). The number of members who grow both vegetables and flowers is 72. Total members is 240. So P(F and V) = 72/240. Step 4: Simplify P(F and V) = 72/240 = 3/10 (dividing numerator and denominator by 24). Step 5: Find P(V). The number of members who grow vegetables is 144. So P(V) = 144/240. Step 6: Simplify P(V) = 144/240 = 3/5 (dividing numerator and denominator by 48). Step 7: Apply the formula: P(F|V) = (3/10) / (3/5) = (3/10) * (5/3) = 15/30 = 1/2. The answer is 1/2.

3. Mere is analyzing the effectiveness of a new reading program at her school. She surveys 400 students and finds that 240 students have improved their reading scores. Of those who improved, 80 also showed increased confidence in reading. Additionally, 40 students showed increased confidence but did not improve their scores. If a randomly selected student from the survey showed increased confidence in reading, what is the probability that they also improved their reading scores? Express your answer as a simplified fraction. Answer: 2/3 Solution: Define the events. Let I be the event that a student improved their reading scores. Total students surveyed: 400 Students who improved (I): 240 Students who improved and showed increased confidence (I and C): 80 Students who showed increased confidence but did not improve: 40 Find the total…
   Full step-by-step solution

   Step 1: Define the events. Let I be the event that a student improved their reading scores. Let C be the event that a student showed increased confidence. Step 2: Identify the given numbers from the problem. Total students surveyed: 400 Students who improved (I): 240 Students who improved and showed increased confidence (I and C): 80 Students who showed increased confidence but did not improve: 40 Step 3: Find the total number of students who showed increased confidence (C). C = (I and C) + (C but not I) = 80 + 40 = 120 Step 4: We want P(I | C), the probability a student improved given they showed increased confidence. Use the conditional probability formula: P(I | C) = P(I and C) / P(C) Step 5: Calculate the probabilities. P(I and C) = 80 / 400 = 1/5 P(C) = 120 / 400 = 3/10 Step 6: Apply the formula. P(I | C) = (1/5) / (3/10) = (1/5) * (10/3) = 10/15 = 2/3 The answer is 2/3.

4. A survey of 200 high school students found that 120 students participate in the science club and 80 students participate in the math club. Among science club members, 30 are also in the math club. If a randomly selected student is in the math club, what is the probability that they are also in the science club? Answer: 0.375 Solution: - Total students: 200 - Science club members: 120 - Math club members: 80 - Students in both clubs: 30 We want P(science club | math club), which means the probability a student is in the science club given that they are in the math club.
   Full step-by-step solution

   Step 1: Identify what we know from the problem: - Total students: 200 - Science club members: 120 - Math club members: 80 - Students in both clubs: 30 Step 2: We want P(science club | math club), which means the probability a student is in the science club given that they are in the math club. Step 3: Use the conditional probability formula: P(A|B) = P(A and B) / P(B) Where A = being in science club, B = being in math club Step 4: Calculate P(A and B) = number in both clubs / total students = 30/200 = 0.15 Step 5: Calculate P(B) = number in math club / total students = 80/200 = 0.4 Step 6: Apply the formula: P(A|B) = P(A and B) / P(B) = 0.15 / 0.4 = 0.375 The answer is 0.375.

5. P(A|B) = P(A∩B)/P(B) where P(A) = 0.6, P(B) = 0.4, P(A∩B) = 0.3 = ? Answer: 0.75 Solution: P(A) = 0.6 P(B) = 0.4 P(A∩B) = 0.3 We want P(A|B). P(A|B) = P(A∩B) / P(B) P(A|B) = 0.3 / 0.4 0.3 divided by 0.4 is the same as 3/4. Convert 3/4 to decimal: 3/4 = 0.75 P(A|B) = 0.75
   Full step-by-step solution

   We are given: P(A) = 0.6 P(B) = 0.4 P(A∩B) = 0.3 We want P(A|B). Step 1: Recall the conditional probability formula: P(A|B) = P(A∩B) / P(B) Step 2: Substitute the known numbers into the formula: P(A|B) = 0.3 / 0.4 Step 3: Perform the division: 0.3 divided by 0.4 is the same as 3/4. Step 4: Convert 3/4 to decimal: 3/4 = 0.75 So the final answer is: P(A|B) = 0.75

6. Aroha runs a diagnostic clinic for a rare plant disease. The test for the disease is 95% accurate at detecting the disease when it is present (true positive rate), but it also has a 4% false positive rate, meaning it incorrectly indicates the disease in 4% of healthy plants. Only 2% of the plants in the region actually have the disease. If a randomly selected plant tests positive for the disease, what is the probability that it actually has the disease? Express your answer as a percentage rounded to two decimal places. Answer: 32.65% Solution: Define events. Let D be the event that a plant has the disease, and T+ be the event that a plant tests positive. P(D) = 0.02 (2% have disease) P(not D) = 0.98 (98% are healthy) P(T+ | D) = 0.95 (true positive rate) P(T+ | not D) = 0.04 (false positive rate) We want P(D | T+), the probability a…
   Full step-by-step solution

   Step 1: Define events. Let D be the event that a plant has the disease, and T+ be the event that a plant tests positive. Step 2: Given probabilities: P(D) = 0.02 (2% have disease) P(not D) = 0.98 (98% are healthy) P(T+ | D) = 0.95 (true positive rate) P(T+ | not D) = 0.04 (false positive rate) Step 3: We want P(D | T+), the probability a plant has the disease given it tested positive. Step 4: Use Bayes' theorem: P(D | T+) = [P(T+ | D) * P(D)] / [P(T+ | D) * P(D) + P(T+ | not D) * P(not D)] Step 5: Calculate the numerator: 0.95 * 0.02 = 0.019 Step 6: Calculate the denominator: (0.95 * 0.02) + (0.04 * 0.98) = 0.019 + 0.0392 = 0.0582 Step 7: P(D | T+) = 0.019 / 0.0582 ≈ 0.32646 Step 8: Convert to percentage: 0.32646 * 100 ≈ 32.65% The answer is 32.65%.

7. P(A|B) = P(A∩B) / P(B) where P(A) = 0.5, P(B) = 0.7, P(A∩B) = 0.35 = ? Answer: 0.5 Solution: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Substitute the given values: P(A|B) = 0.35 / 0.7 Perform the division: 0.35 ÷ 0.7 = 0.5 The conditional probability P(A|B) equals 0.5
   Full step-by-step solution

   Step 1: Write the conditional probability formula: P(A|B) = P(A∩B) / P(B) Step 2: Substitute the given values: P(A|B) = 0.35 / 0.7 Step 3: Perform the division: 0.35 ÷ 0.7 = 0.5 Step 4: The conditional probability P(A|B) equals 0.5