Probability Concepts

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Probability Concepts · Grade 7 · Worksheet 3

1. Liam is conducting a probability experiment with a standard six-sided die and a fair coin. He first rolls the die and then flips the coin. What is the probability that he rolls a number greater than 4 and the coin lands on tails? Answer: 1/6 Solution: Identify the two independent events. - Event A: Rolling a number greater than 4 on a six-sided die. - Event B: The coin landing on tails.
   Full step-by-step solution

   Let's solve this step by step. Step 1: Identify the two independent events. - Event A: Rolling a number greater than 4 on a six-sided die. - Event B: The coin landing on tails. Step 2: Find the probability of Event A. A standard die has faces numbered 1, 2, 3, 4, 5, 6. Numbers greater than 4 are 5 and 6. So there are 2 favorable outcomes out of 6 possible outcomes. Probability(A) = 2/6 = 1/3. Step 3: Find the probability of Event B. A fair coin has two equally likely outcomes: heads or tails. Favorable outcome: tails (1 outcome). Probability(B) = 1/2. Step 4: Combine the probabilities. Since the die roll and coin flip are independent, the probability of both happening is the product of their individual probabilities. Probability(A and B) = Probability(A) × Probability(B) = (1/3) × (1/2). Step 5: Multiply the fractions. (1/3) × (1/2) = 1/6. Step 6: Conclusion. The probability that Liam rolls a number greater than 4 and the coin lands on tails is 1/6. Final answer: 1/6

2. P(drawing a card numbered with a multiple of 7 from a bag containing cards numbered 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98) = ? Answer: 1 Solution: Count the total number of cards in the bag. The cards are numbered 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98. That is 14 cards total.
   Full step-by-step solution

   Step 1: Count the total number of cards in the bag. The cards are numbered 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98. That is 14 cards total. Step 2: Identify the favorable outcomes. The problem asks for drawing a card with a multiple of 7. All numbers listed are multiples of 7, so all 14 cards are favorable. Step 3: Calculate the probability. Probability = (Number of favorable outcomes) / (Total number of outcomes) = 14/14. Step 4: Simplify the fraction. 14/14 = 1. The answer is 1.

3. 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.

4. P(rolling a 3 or 4 on a fair six-sided die) = ? Answer: 1/3 Solution: We are rolling a fair six-sided die. The faces are numbered 1, 2, 3, 4, 5, and 6. We want the probability of rolling a 3 OR a 4.
   Full step-by-step solution

   Step 1: Understand the problem. We are rolling a fair six-sided die. The faces are numbered 1, 2, 3, 4, 5, and 6. We want the probability of rolling a 3 OR a 4. Step 2: Recall the basic probability formula. For an event, Probability = (Number of favorable outcomes) / (Total number of possible outcomes). Step 3: Identify the total number of possible outcomes. Since the die is fair and has six sides, there are 6 possible outcomes (1, 2, 3, 4, 5, 6). So, total outcomes = 6. Step 4: Identify the number of favorable outcomes. The favorable outcomes are the outcomes we want: rolling a 3 or rolling a 4. This gives us 2 favorable outcomes (3 and 4). Step 5: Apply the probability formula. Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = 2 / 6. Step 6: Simplify the fraction. The fraction 2/6 can be simplified by dividing the numerator and denominator by 2. 2 divided by 2 is 1, and 6 divided by 2 is 3. So, 2/6 simplifies to 1/3. Step 7: State the final answer. Therefore, the probability of rolling a 3 or a 4 is 1/3.

5. P(drawing a card numbered with a multiple of 7 from a bag containing cards numbered 7, 14, 21, 28, 35, 42, 49, 56, 63, 70) = ? Answer: 1 Solution: Count the total number of cards in the bag. The cards are numbered 7, 14, 21, 28, 35, 42, 49, 56, 63, 70. That is 10 cards total.
   Full step-by-step solution

   Step 1: Count the total number of cards in the bag. The cards are numbered 7, 14, 21, 28, 35, 42, 49, 56, 63, 70. That is 10 cards total. Step 2: Identify the favorable outcomes. The problem asks for drawing a card with a multiple of 7. All numbers listed (7, 14, 21, 28, 35, 42, 49, 56, 63, 70) are multiples of 7. So there are 10 favorable outcomes. Step 3: Calculate the probability. Probability = (Number of favorable outcomes) / (Total number of outcomes) = 10/10. Step 4: Simplify the fraction. 10/10 = 1. The answer is 1.

6. Emma is playing a board game where she needs to roll two fair six-sided dice and get a sum of 7 or 11 to win on her turn. What is the probability that Emma wins on her next turn? Express your answer as a simplified fraction. Answer: 2/9 Solution: Determine the total number of possible outcomes when rolling two dice. Each die has 6 faces, so total outcomes = 6 × 6 = 36. Identify all outcomes that give a sum of 7.
   Full step-by-step solution

   Step 1: Determine the total number of possible outcomes when rolling two dice. Each die has 6 faces, so total outcomes = 6 × 6 = 36. Step 2: Identify all outcomes that give a sum of 7. These are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes Step 3: Identify all outcomes that give a sum of 11. These are: (5,6), (6,5) = 2 outcomes Step 4: Calculate total favorable outcomes. Favorable outcomes = outcomes with sum 7 + outcomes with sum 11 = 6 + 2 = 8 Step 5: Calculate the probability. Probability = favorable outcomes / total outcomes = 8/36 Step 6: Simplify the fraction. 8/36 = 2/9 The answer is 2/9.

7. P(rolling an even number or a number greater than 4 on a fair six-sided die) = ? Answer: 2/3 Solution: List the sample space: {1, 2, 3, 4, 5, 6} Identify even numbers: {2, 4, 6} Identify numbers greater than 4: {5, 6} Combine the favorable outcomes: {2, 4, 5, 6} Count the favorable outcomes: 4 Count the total possible outcomes: 6 Calculate the probability: 4/6 Simplify the fraction: 4/6 = 2/3 The…
   Full step-by-step solution

   Step 1: List the sample space: {1, 2, 3, 4, 5, 6} Step 2: Identify even numbers: {2, 4, 6} Step 3: Identify numbers greater than 4: {5, 6} Step 4: Combine the favorable outcomes: {2, 4, 5, 6} Step 5: Count the favorable outcomes: 4 Step 6: Count the total possible outcomes: 6 Step 7: Calculate the probability: 4/6 Step 8: Simplify the fraction: 4/6 = 2/3 The answer is 2/3.

8. P(rolling a number divisible by 3 on a fair twelve-sided die) = ? Answer: 1/3 Solution: A fair twelve-sided die has numbers 1 through 12, so there are 12 total possible outcomes. Numbers divisible by 3 between 1 and 12 are: 3, 6, 9, 12. That's 4 favorable outcomes.
   Full step-by-step solution

   Step 1: A fair twelve-sided die has numbers 1 through 12, so there are 12 total possible outcomes. Step 2: Numbers divisible by 3 between 1 and 12 are: 3, 6, 9, 12. That's 4 favorable outcomes. Step 3: Probability = (favorable outcomes) / (total outcomes) = 4/12 Step 4: Simplify the fraction: 4/12 = 1/3 The answer is 1/3.

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

   Step 1: List all possible outcomes when rolling a fair six-sided die: {1, 2, 3, 4, 5, 6}. There are 6 total outcomes. Step 2: Identify the favorable outcomes for rolling a 2: {2}. Step 3: Identify the favorable outcomes for rolling an even number: {2, 4, 6}. Step 4: Combine the favorable outcomes for '2 or even': {2, 4, 6}. The number 2 appears in both sets but is counted only once. Step 5: Count the number of favorable outcomes: There are 3 outcomes (2, 4, 6). Step 6: Calculate the probability: Number of favorable outcomes / Total number of outcomes = 3/6. Step 7: Simplify the fraction: 3/6 = 1/2. The answer is 1/2.