Probability Concepts

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Probability Concepts · Grade 7 · Worksheet 2

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

   Step 1: Understand the problem. We have a fair six-sided die. The possible outcomes when rolling it are: 1, 2, 3, 4, 5, 6. We want the probability of rolling a 3 OR a 5. Step 2: Recall the formula for probability. Probability = (Number of favorable outcomes) / (Total number of possible outcomes). Step 3: Identify the total number of possible outcomes. Since the die has six sides, there are 6 possible outcomes. So, total outcomes = 6. Step 4: Identify the number of favorable outcomes. Favorable outcomes are the outcomes we want, which are rolling a 3 OR rolling a 5. - Rolling a 3 is one favorable outcome. - Rolling a 5 is another favorable outcome. So, total favorable outcomes = 1 + 1 = 2. Step 5: Calculate the probability. 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. 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 5 is 1/3.

2. P(drawing a card numbered with an odd number from a bag containing cards numbered 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21) = ? Answer: 1 Solution: Count the total number of cards in the bag. The cards are numbered 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. That is 11 cards total.
   Full step-by-step solution

   Step 1: Count the total number of cards in the bag. The cards are numbered 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. That is 11 cards total. Step 2: Identify the favorable outcomes. The problem asks for drawing a card with an odd number. All numbers listed are odd, so all 11 cards are favorable. Step 3: Calculate the probability. Probability = (Number of favorable outcomes) / (Total number of outcomes) = 11/11. Step 4: Simplify the fraction. 11/11 = 1. The answer is 1.

3. A circular dartboard has a radius of 18 inches. The bullseye is a smaller circle in the center with a radius of 3 inches. If a dart hits the dartboard at a completely random point, what is the probability that it lands in the bullseye? Express your answer as a fraction in simplest form. Answer: 1/36 Solution: We have a circular dartboard with radius 18 inches. The bullseye is a smaller circle in the center with radius 3 inches. A dart hits a random point on the dartboard.
   Full step-by-step solution

   Step 1: Understand the problem. We have a circular dartboard with radius 18 inches. The bullseye is a smaller circle in the center with radius 3 inches. A dart hits a random point on the dartboard. We want the probability that it lands in the bullseye. Step 2: Recall probability for areas. When points are chosen randomly over an area, the probability of hitting a specific region is: (Area of the specific region) / (Total area available) Step 3: Find the area of the whole dartboard. The area of a circle is pi * r^2. Total area = pi * (18)^2 = pi * 324. Step 4: Find the area of the bullseye. Bullseye area = pi * (3)^2 = pi * 9. Step 5: Write the probability. Probability = (Bullseye area) / (Total area) = (pi * 9) / (pi * 324). Step 6: Simplify. The pi cancels out: 9 / 324. Step 7: Reduce the fraction. Both 9 and 324 are divisible by 9: 9 ÷ 9 = 1 324 ÷ 9 = 36 So, 9/324 = 1/36. Step 8: Final answer. The probability is 1/36.

4. A circular dartboard has a radius of 12 inches. The bullseye is a smaller circle at the center with a radius of 2 inches. If a randomly thrown dart hits the dartboard, what is the probability that it lands in the bullseye? (Use π = 3.14) Answer: 0.0278 Solution: We have a circular dartboard with radius 12 inches. The bullseye is a smaller circle at the center with radius 2 inches.
   Full step-by-step solution

   Step 1: Understand the problem We have a circular dartboard with radius 12 inches. The bullseye is a smaller circle at the center with radius 2 inches. A dart hits the dartboard randomly, so the probability of hitting the bullseye is the ratio of the area of the bullseye to the area of the whole dartboard. Step 2: Recall the area formula for a circle Area of a circle = π × r² Step 3: Calculate the area of the dartboard Radius of dartboard, R = 12 inches Area of dartboard = π × R² = 3.14 × (12)² First compute 12² = 144 So Area of dartboard = 3.14 × 144 3.14 × 144 = 3.14 × 100 + 3.14 × 44 = 314 + 138.16 = 452.16 square inches Step 4: Calculate the area of the bullseye Radius of bullseye, r = 2 inches Area of bullseye = π × r² = 3.14 × (2)² 2² = 4 So Area of bullseye = 3.14 × 4 = 12.56 square inches Step 5: Compute the probability Probability = (Area of bullseye) / (Area of dartboard) = 12.56 / 452.16 Step 6: Simplify the division First, let's divide numerator and denominator by 0.01 to make it easier: 1256 / 45216 Now divide both by 8: 1256 ÷ 8 = 157 45216 ÷ 8 = 5652 So we have 157 / 5652 Step 7: Perform the division numerically 12.56 ÷ 452.16 We can compute: 452.16 × 0.027 = 12.20832 12.56 − 12.20832 = 0.35168 Now 452.16 × 0.0007 = 0.316512 0.35168 − 0.316512 = 0.035168 452.16 × 0.000077 ≈ 0.034816 Remainder is small, so approximately 0.027777... Step 8: Round to four decimal places 0.027777... ≈ 0.0278 Step 9: Final answer The probability that the dart lands in the bullseye is 0.0278.

5. Noah is organizing a school raffle where 150 tickets are sold. There are 3 grand prizes, 8 second prizes, and 15 consolation prizes. If Noah buys one ticket, what is the probability that he wins any prize? Express your answer as a simplified fraction. Answer: 13/75 Solution: Grand prizes: 3 tickets Second prizes: 8 tickets Consolation prizes: 15 tickets Total prize tickets = 3 + 8 + 15 = 26 Total tickets sold = 150 Probability = (Number of favorable outcomes) / (Total possible outcomes) Probability = 26/150 Both numerator and denominator can be divided by 2 26 ÷ 2 =…
   Full step-by-step solution

   Step 1: Find the total number of prize tickets Grand prizes: 3 tickets Second prizes: 8 tickets Consolation prizes: 15 tickets Total prize tickets = 3 + 8 + 15 = 26 Step 2: Find the total number of tickets Total tickets sold = 150 Step 3: Calculate the probability Probability = (Number of favorable outcomes) / (Total possible outcomes) Probability = 26/150 Step 4: Simplify the fraction Both numerator and denominator can be divided by 2 26 ÷ 2 = 13 150 ÷ 2 = 75 Simplified fraction = 13/75 The answer is 13/75.