Exercises
Statistics I
Week 7 Graded Assignment

Week 7 Graded Assignment


Statistics Graded Assignment

Questions and Solutions

1. Arranging Boys and Girls with Exactly 4 Boys Between Two Girls

Question: $ m $ boys and 2 girls are to be placed next to each other in the school ground for morning assembly. What is the probability that there are exactly 4 boys between the 2 girls?

Options: a. $\frac{2 m-5}{m+2 P_{2}}$ b. $\frac{2 m-6}{m+2 P_{2}}$ c. $\frac{2 m-6}{m+3 P_{2}}$ d. $\frac{2 m-4}{m+2 P_{2}}$

Answer: b

Solution:

  • Total arrangements for 2 girls: $^{m+2}P_2$
  • Favorable cases: For exactly 4 boys between two girls, there are $2(m-3)$ ways (since for each starting position, there are two orders for the girls).
  • Probability: $\frac{2m-6}{^{m+2}P_2}$

2. Cardinality of Event in Multiple Select Question

Question: In a Multiple Select Question, there are $ m $ options, of which one or more can be correct. Let us define an event $ E $ that the option ‘A’ is correct. What is the cardinality of $ E $?

Solution:

  • Case 1: Only option A is correct.
  • Case 2: Two options are correct and A is one of them.
  • …
  • Case m: All options are correct.
  • Total: $1 + ^{m-1}C_1 + ^{m-1}C_2 + \ldots + 1 = 2^{m-1}$

3. Probability of At Least Two Correct Predictions

Question: A person predicts daily whether the price of stocks of wrist watch companies will go up or down. If his prediction on stock price of Titan is correct $ a $ times out of $ b $, for Rolex it is correct $ p $ times out of $ q $ and for Fossil it is correct $ x $ times out of $ y $, then what is the probability that at least two of his predictions are correct on a given day?

Options: a. $\left[\frac{a}{b} \times \frac{p}{q} \times\left(1-\frac{x}{y}\right)\right]+\left[\frac{a}{b} \times\left(1-\frac{p}{q}\right) \times \frac{x}{y}\right]+\left[\left(1-\frac{a}{b}\right) \times \frac{p}{q} \times \frac{x}{y}\right]+\left[\frac{a}{b} \times \frac{p}{q} \times \frac{x}{y}\right]$ b. $\left[\frac{a}{b} \times \frac{p}{q} \times\left(1-\frac{x}{y}\right)\right]+\left[\frac{a}{b} \times\left(1-\frac{p}{q}\right) \times \frac{x}{y}\right]+\left[\left(1-\frac{a}{b}\right) \times \frac{p}{q} \times \frac{x}{y}\right]$ c. $\left[\frac{a}{b} \times \frac{p}{q} \times \frac{x}{y}\right]+\left[\frac{a}{b} \times \frac{p}{q} \times \frac{x}{y}\right]+\left[\frac{a}{b} \times \frac{p}{q} \times \frac{x}{y}\right]+\left[\frac{a}{b} \times \frac{p}{q} \times \frac{x}{y}\right]$ d. $\left[\frac{a}{b} \times \frac{p}{q} \times \frac{x}{y}\right]+\left[\frac{a}{b} \times \frac{p}{q} \times \frac{x}{y}\right]+\left[\frac{a}{b} \times \frac{p}{q} \times \frac{x}{y}\right]$

Answer: a

Solution:

  • Case 1: Only Titan and Rolex correct.
  • Case 2: Only Titan and Fossil correct.
  • Case 3: Only Rolex and Fossil correct.
  • Case 4: All three correct.
  • Probability: Sum of all four cases.

4. Probability of Being in Table Tennis Team Only

Question: There are a total of $ n $ students who are part of badminton, table tennis and tennis team of the college. Of which $ x% $ of students play table tennis, $ p% $ play tennis and $ a% $ play badminton. It is also noticed that all students who play tennis also play table tennis, but not badminton. Now a student is selected at random, what is the probability that he/she is part of table tennis team only? (Enter the answer correct to 1 decimal place.)

Solution:

  • Probability: $\left(1 - \frac{a + p}{100}\right)$ (assuming all students are accounted for and $x% \geq p% + a%$).
  • Example: $ n = 150, x = 50, p = 20, a = 70 $
    • Table tennis only: $\frac{15}{150} = \frac{1}{10}$

5. Probability of Clearing Quiz 2

Question: The chance that a student will clear the quiz 1 paper is $ a $ and the chance that he will clear both quiz 1 and quiz 2 papers is $ b $. The chance that he will clear at least one quiz paper is $ c $. What is the chance that he will clear quiz 2 paper? (Enter the answer correct to 2 decimal accuracy)

Solution:

  • Given: $ P(A) = a, P(A \cap B) = b, P(A \cup B) = c $
  • Find: $ P(B) $
  • Formula: $ P(B) = c - a + b $
  • Example: $ a = 0.4, b = 0.3, c = 0.5 $
    • $ P(B) = 0.5 - 0.4 + 0.3 = 0.4 $

6. Probability of Union of Two Events

Question: If $ P(A) = x $ and $ P(B) = y $ and probability of the complement of $(A \cup B)$ is $ z $, then calculate $ P(A \cup B) $? (Enter the answer correct to 2 decimal point accuracy)

Solution:

  • Formula: $ P(A \cup B) = 1 - z $
  • Example: $ x = 0.2, y = 0.5, z = 0.4 $
    • $ P(A \cup B) = 1 - 0.4 = 0.6 $

7. Probability of Drawing Specific Cards

Question: $ a $ cards are drawn at random (without replacement) from a pack of 52 cards. Find the probability that $ b $ are black and $ c $ are red. (Enter the answer correct to two decimal places)

Solution:

  • Probability: $\frac{^{26}C_b \times ^{26}C_c}{^{52}C_a}$
  • Example: $ a = 4, b = 2, c = 2 $
    • Probability: $\frac{^{26}C_2 \times ^{26}C_2}{^{52}C_4} \approx 0.39$

8–10: Clothes Selection Probability (Common Data)

Pramod goes to a shop to buy some clothes. Shopkeeper shows him $ x $ shirts, $ y $ pants and $ z $ t-shirts. If he selects three clothes at random, then based on the information, answer the following:

8. Probability of Different Types

Question: Find the probability that the randomly chosen clothes are of different type.

Solution:

  • Total cases: $^{x+y+z}C_3$
  • Favorable cases: $^xC_1 \times ^yC_1 \times ^zC_1$
  • Probability: $\frac{^xC_1 \times ^yC_1 \times ^zC_1}{^{x+y+z}C_3}$
  • Example: $ x=5, y=4, z=10 $
    • Probability: $\frac{5 \times 4 \times 10}{^{19}C_3} \approx 0.206$

9. Probability of No Pants

Question: Find the probability that the randomly chosen clothes does not contain pant.

Solution:

  • Favorable cases: $^{x+z}C_3$
  • Probability: $\frac{^{x+z}C_3}{^{x+y+z}C_3}$
  • Example: $ x=5, y=4, z=10 $
    • Probability: $\frac{^{15}C_3}{^{19}C_3} \approx 0.469$

10. Probability of At Least One Shirt

Question: Find the probability that at least one of the clothes is a shirt.

Solution:

  • Probability: $1 - \frac{^{y+z}C_3}{^{x+y+z}C_3}$
  • Example: $ x=5, y=4, z=10 $
    • Probability: $1 - \frac{^{14}C_3}{^{19}C_3} \approx 0.624$

11. Probability of Imaginary Roots

Question: An urn contains 3 balls numbered 1, 2 and 3. The coefficients of the equation $ p x^2 + q x + c = 0 $ are determined by drawing the numbered balls with replacement. What is the probability that the equation will have imaginary roots?

Options: a. $\frac{4}{27}$ b. $\frac{23}{27}$ c. $\frac{16}{27}$ d. None of the above

Answer: b

Solution:

  • Total outcomes: $3 \times 3 \times 3 = 27$
  • Favorable for real roots: Only 4 cases (see solution in PDF)
  • Probability of imaginary roots: $1 - \frac{4}{27} = \frac{23}{27}$

12. Mutually Exclusive Events

Question: If $ A $ and $ B $ are mutually exclusive or disjoint events, then which of the following is/are always true: a. $ P(A) = P(B) $ b. $ P(A) < P(B) $ c. $ P(A) \leq P(B^c) $ d. $ P(A) \geq P(B^c) $

Answer: c

Solution: Since $ A \cap B = \emptyset $, $ A \subseteq B^c $, so $ P(A) \leq P(B^c) $.

This covers all questions and solutions from the Week_7_Graded_Solution.pdf1.

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  1. Week_7_Graded_Solution.pdf ↩︎

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