Probability Concepts

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Probability Concepts · Grade 7 · Worksheet 1

1. Emma is designing a board game that uses a spinner divided into 12 equal sections. The sections are colored: 4 red, 3 blue, 3 green, and 2 yellow. She also uses a standard six-sided die numbered 1-6. If a player spins the spinner and then rolls the die, what is the probability that they land on a green section AND roll a prime number? Express your answer as a simplified fraction. Answer: 1/8 Solution: Find the probability of landing on green on the spinner. There are 3 green sections out of 12 total sections. Probability of green = 3/12 = 1/4 Find the probability of rolling a prime number on the die.
   Full step-by-step solution

   Step 1: Find the probability of landing on green on the spinner. There are 3 green sections out of 12 total sections. Probability of green = 3/12 = 1/4 Step 2: Find the probability of rolling a prime number on the die. The prime numbers on a standard die are 2, 3, and 5. There are 3 prime numbers out of 6 possible outcomes. Probability of prime number = 3/6 = 1/2 Step 3: Since these are independent events, multiply the probabilities. Probability of both events = (1/4) × (1/2) = 1/8 The answer is 1/8.

2. P(drawing a card numbered with a multiple of 9 from a bag containing cards numbered 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135) = ? Answer: 1 Solution: Count the total number of cards in the bag. The cards are numbered: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135. That is 15 cards total.
   Full step-by-step solution

   Step 1: Count the total number of cards in the bag. The cards are numbered: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135. That is 15 cards total. Step 2: Identify the favorable outcomes. The problem asks for drawing a card with a multiple of 9. All numbers listed (9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135) are multiples of 9. So there are 15 favorable outcomes. Step 3: Calculate the probability. Probability = (Number of favorable outcomes) / (Total number of outcomes) = 15/15. Step 4: Simplify the fraction. 15/15 = 1. The answer is 1.

3. A factory produces 15,000 electronic components per day. Quality control testing shows that 2.5% of components are defective. If a customer orders 8,000 components, what is the expected number of defective components in their order? Round your answer to the nearest whole number. Answer: 200 Solution: The factory produces components with a defect rate of 2.5%. This means that for any given component, the probability of it being defective is 2.5%.
   Full step-by-step solution

   Step 1: Understand the problem. The factory produces components with a defect rate of 2.5%. This means that for any given component, the probability of it being defective is 2.5%. If a customer orders 8,000 components, the expected number of defective components is found by multiplying the total ordered by the defect rate. Step 2: Convert the percentage to a decimal. 2.5% = 2.5 / 100 = 0.025 Step 3: Multiply the total order quantity by the defect rate. Expected defective = 8,000 × 0.025 Step 4: Perform the multiplication. 8,000 × 0.025 = 200 Step 5: Round to the nearest whole number. 200 is already a whole number. Step 6: Final answer. The expected number of defective components in the order is 200.

4. P(rolling a 1 or 6 on a fair six-sided die) = ? Answer: 1/3 Solution: Identify the total number of possible outcomes when rolling a fair six-sided die. There are 6 possible outcomes (1, 2, 3, 4, 5, 6). Identify the number of favorable outcomes (rolling a 1 or a 6).
   Full step-by-step solution

   Step 1: Identify the total number of possible outcomes when rolling a fair six-sided die. There are 6 possible outcomes (1, 2, 3, 4, 5, 6). Step 2: Identify the number of favorable outcomes (rolling a 1 or a 6). There are 2 favorable outcomes. Step 3: Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes: 2/6. Step 4: Simplify the fraction 2/6 by dividing the numerator and denominator by 2: 2/6 = 1/3. The answer is 1/3.

5. P(drawing a card numbered with a multiple of 6 from a bag containing cards numbered 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96) = ? Answer: 1 Solution: Count the total number of cards in the bag. The cards are numbered 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96. That is 16 cards total.
   Full step-by-step solution

   Step 1: Count the total number of cards in the bag. The cards are numbered 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96. That is 16 cards total. Step 2: Identify the favorable outcomes. The problem asks for drawing a card with a multiple of 6. All numbers listed are multiples of 6, so all 16 cards are favorable. Step 3: Calculate the probability. Probability = (Number of favorable outcomes) / (Total number of outcomes) = 16/16. Step 4: Simplify the fraction. 16/16 = 1. The answer is 1.

6. P(drawing a heart or a face card from a standard deck) = ? Answer: 11/26 Solution: A standard deck has 52 cards. There are 13 hearts in the deck. There are 12 face cards in the deck (Jack, Queen, King in each of the 4 suits).
   Full step-by-step solution

   Step 1: A standard deck has 52 cards. Step 2: There are 13 hearts in the deck. Step 3: There are 12 face cards in the deck (Jack, Queen, King in each of the 4 suits). Step 4: Some cards are both hearts AND face cards. There are 3 of these (Jack of Hearts, Queen of Hearts, King of Hearts). Step 5: Use the formula for P(A or B) = P(A) + P(B) - P(A and B). Step 6: P(heart or face card) = (13/52) + (12/52) - (3/52) = (13 + 12 - 3)/52 = 22/52. Step 7: Simplify the fraction: 22/52 = 11/26. The answer is 11/26.

7. Emma is conducting a survey about favorite school subjects in her grade 7 class of 32 students. She randomly selects 2 students from the class to interview. If there are 18 girls and 14 boys in the class, what is the probability that both selected students will be boys? Express your answer as a simplified fraction. Answer: 91/496 Solution: Calculate the probability that the first student selected is a boy. There are 14 boys out of 32 total students, so P(first boy) = 14/32 = 7/16 Calculate the probability that the second student selected is a boy, given that the first was a boy.
   Full step-by-step solution

   Step 1: Calculate the probability that the first student selected is a boy. There are 14 boys out of 32 total students, so P(first boy) = 14/32 = 7/16 Step 2: Calculate the probability that the second student selected is a boy, given that the first was a boy. After selecting one boy, there are now 13 boys left out of 31 total students, so P(second boy | first boy) = 13/31 Step 3: Multiply the probabilities to find the probability that both are boys. P(both boys) = (7/16) × (13/31) = 91/496 The answer is 91/496.

8. P(rolling a prime number on a fair six-sided die) = ? Answer: 1/2 Solution: Identify the possible outcomes when rolling a fair six-sided die. The faces are numbered 1, 2, 3, 4, 5, 6. Define what a prime number is.
   Full step-by-step solution

   Step 1: Identify the possible outcomes when rolling a fair six-sided die. The faces are numbered 1, 2, 3, 4, 5, 6. So, the total number of possible outcomes is 6. Step 2: Define what a prime number is. A prime number is a whole number greater than 1 whose only factors are 1 and itself. Step 3: Check each number on the die to see if it is prime. - Check 1: 1 is not greater than 1, so it is not prime. - Check 2: 2 is greater than 1, and its only factors are 1 and 2. So, 2 is prime. - Check 3: 3 is greater than 1, and its only factors are 1 and 3. So, 3 is prime. - Check 4: 4 is greater than 1, but its factors are 1, 2, and 4. Since it has a factor other than 1 and itself (2), it is not prime. - Check 5: 5 is greater than 1, and its only factors are 1 and 5. So, 5 is prime. - Check 6: 6 is greater than 1, but its factors are 1, 2, 3, and 6. Since it has factors other than 1 and itself (2 and 3), it is not prime. Step 4: List the prime numbers from the die. The prime numbers are 2, 3, and 5. So, the number of favorable outcomes (rolling a prime) is 3. Step 5: Calculate the probability. Probability is given by (Number of favorable outcomes) / (Total number of possible outcomes). So, P(prime) = 3 / 6. Step 6: Simplify the fraction. 3/6 simplifies to 1/2 by dividing both the numerator and denominator by 3. Therefore, the probability of rolling a prime number is 1/2.