Statistics. Level 7. Probability. Sample Space Diagrams | PDF - Free Printable
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Step-by-step solution for: Statistics. Level 7. Probability. Sample Space Diagrams | PDF
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Step-by-step solution for: Statistics. Level 7. Probability. Sample Space Diagrams | PDF
1) Jamie has two fair spinners. One has three equal sections coloured red, blue and yellow. The other has four equal sections numbered 1, 2, 3 and 4. Jamie spins both spinners.
Complete the sample space diagram.
- The sample space diagram will have the colours (Red, Blue, Yellow) on one axis and the numbers (1, 2, 3, 4) on the other axis.
- The possible outcomes are:
- Red with 1, Red with 2, Red with 3, Red with 4
- Blue with 1, Blue with 2, Blue with 3, Blue with 4
- Yellow with 1, Yellow with 2, Yellow with 3, Yellow with 4
- There are 12 possible outcomes in total.
2) A red dice and a blue dice are thrown together and the total of the two scores is found. The red dice is numbered 1 to 6. The blue dice is numbered 1, 2, 3, 3, 5, 8.
a. Complete the sample space diagram.
- The sample space diagram will have the red dice outcomes (1 to 6) on one axis and the blue dice outcomes (1, 2, 3, 3, 5, 8) on the other axis.
- The total scores will be the sum of the numbers on the red and blue dice.
- The possible totals range from 2 (1+1) to 14 (6+8).
b. Which are the most likely scores?
- The most likely scores are those that occur most frequently.
- The score 4 can be achieved in 3 ways: (1,3), (2,2), (3,1) — but note the blue dice has two 3s, so (1,3) occurs twice, (2,2) once, (3,1) once, totaling 4 ways.
- The score 5 can be achieved in 3 ways: (2,3), (3,2), (4,1) — (2,3) occurs twice, (3,2) once, (4,1) once, totaling 4 ways.
- The score 6 can be achieved in 3 ways: (3,3), (4,2), (5,1) — (3,3) occurs twice, (4,2) once, (5,1) once, totaling 4 ways.
- The score 7 can be achieved in 3 ways: (4,3), (5,2), (6,1) — (4,3) occurs twice, (5,2) once, (6,1) once, totaling 4 ways.
- The score 8 can be achieved in 3 ways: (5,3), (6,2) — (5,3) occurs twice, (6,2) once, totaling 3 ways.
- The score 9 can be achieved in 2 ways: (6,3) — (6,3) occurs twice, totaling 2 ways.
- The score 10 can be achieved in 1 way: (2,8) — (2,8) once, (3,8) once, (4,8) once, (5,8) once, (6,8) once — totaling 5 ways.
- Wait, let's recalculate:
- For 10: (2,8), (3,8), (4,8), (5,8), (6,8) — 5 ways.
- For 11: (3,8), (4,8), (5,8), (6,8) — 4 ways.
- For 12: (4,8), (5,8), (6,8) — 3 ways.
- For 13: (5,8), (6,8) — 2 ways.
- For 14: (6,8) — 1 way.
- The most likely score is 10, which occurs 5 times.
c. Which scores are least likely?
- The least likely scores are those that occur only once.
- The scores 2, 3, 13, and 14 occur only once.
- So, the least likely scores are 2, 3, 13, and 14.
d. Find the probability that the total score is less than 9.
- The total score is less than 9 means the total is 2, 3, 4, 5, 6, 7, or 8.
- The number of ways to get each score:
- 2: (1,1) — 1 way
- 3: (1,2), (2,1) — 2 ways
- 4: (1,3), (2,2), (3,1) — (1,3) occurs twice, (2,2) once, (3,1) once — 4 ways
- 5: (2,3), (3,2), (4,1) — (2,3) occurs twice, (3,2) once, (4,1) once — 4 ways
- 6: (3,3), (4,2), (5,1) — (3,3) occurs twice, (4,2) once, (5,1) once — 4 ways
- 7: (4,3), (5,2), (6,1) — (4,3) occurs twice, (5,2) once, (6,1) once — 4 ways
- 8: (5,3), (6,2) — (5,3) occurs twice, (6,2) once — 3 ways
- Total ways: 1 + 2 + 4 + 4 + 4 + 4 + 3 = 22 ways.
- Total possible outcomes: 6 (red) * 6 (blue) = 36.
- Probability = 22/36 = 11/18.
4) A fair dice and a coin are thrown together. Using a sample space diagram:
a. Work out how many possible outcomes there are.
- The dice has 6 outcomes (1 to 6).
- The coin has 2 outcomes (Heads, Tails).
- Total outcomes: 6 * 2 = 12.
b. Work out the probability of getting an even number and a tails.
- Even numbers on dice: 2, 4, 6 — 3 outcomes.
- Tails: 1 outcome.
- Favorable outcomes: (2,T), (4,T), (6,T) — 3 outcomes.
- Probability = 3/12 = 1/4.
5) The royal cards of clubs and the royal cards of diamonds are randomly put into two piles. A card is taken from each pile, work out the probability that:
a. Both cards will be a Queen.
- There are 4 royal cards: King, Queen, Jack, Ace for each suit.
- So, 2 queens (one from clubs, one from diamonds).
- The probability that the first card is a Queen from clubs is 1/4.
- The probability that the second card is a Queen from diamonds is 1/4.
- Since the draws are independent, the probability that both are Queens is (1/4) * (1/4) = 1/16.
b. At least one card will be a Jack.
- The probability that the first card is not a Jack is 3/4.
- The probability that the second card is not a Jack is 3/4.
- The probability that neither is a Jack is (3/4) * (3/4) = 9/16.
- The probability that at least one is a Jack is 1 - 9/16 = 7/16.
Complete the sample space diagram.
- The sample space diagram will have the colours (Red, Blue, Yellow) on one axis and the numbers (1, 2, 3, 4) on the other axis.
- The possible outcomes are:
- Red with 1, Red with 2, Red with 3, Red with 4
- Blue with 1, Blue with 2, Blue with 3, Blue with 4
- Yellow with 1, Yellow with 2, Yellow with 3, Yellow with 4
- There are 12 possible outcomes in total.
2) A red dice and a blue dice are thrown together and the total of the two scores is found. The red dice is numbered 1 to 6. The blue dice is numbered 1, 2, 3, 3, 5, 8.
a. Complete the sample space diagram.
- The sample space diagram will have the red dice outcomes (1 to 6) on one axis and the blue dice outcomes (1, 2, 3, 3, 5, 8) on the other axis.
- The total scores will be the sum of the numbers on the red and blue dice.
- The possible totals range from 2 (1+1) to 14 (6+8).
b. Which are the most likely scores?
- The most likely scores are those that occur most frequently.
- The score 4 can be achieved in 3 ways: (1,3), (2,2), (3,1) — but note the blue dice has two 3s, so (1,3) occurs twice, (2,2) once, (3,1) once, totaling 4 ways.
- The score 5 can be achieved in 3 ways: (2,3), (3,2), (4,1) — (2,3) occurs twice, (3,2) once, (4,1) once, totaling 4 ways.
- The score 6 can be achieved in 3 ways: (3,3), (4,2), (5,1) — (3,3) occurs twice, (4,2) once, (5,1) once, totaling 4 ways.
- The score 7 can be achieved in 3 ways: (4,3), (5,2), (6,1) — (4,3) occurs twice, (5,2) once, (6,1) once, totaling 4 ways.
- The score 8 can be achieved in 3 ways: (5,3), (6,2) — (5,3) occurs twice, (6,2) once, totaling 3 ways.
- The score 9 can be achieved in 2 ways: (6,3) — (6,3) occurs twice, totaling 2 ways.
- The score 10 can be achieved in 1 way: (2,8) — (2,8) once, (3,8) once, (4,8) once, (5,8) once, (6,8) once — totaling 5 ways.
- Wait, let's recalculate:
- For 10: (2,8), (3,8), (4,8), (5,8), (6,8) — 5 ways.
- For 11: (3,8), (4,8), (5,8), (6,8) — 4 ways.
- For 12: (4,8), (5,8), (6,8) — 3 ways.
- For 13: (5,8), (6,8) — 2 ways.
- For 14: (6,8) — 1 way.
- The most likely score is 10, which occurs 5 times.
c. Which scores are least likely?
- The least likely scores are those that occur only once.
- The scores 2, 3, 13, and 14 occur only once.
- So, the least likely scores are 2, 3, 13, and 14.
d. Find the probability that the total score is less than 9.
- The total score is less than 9 means the total is 2, 3, 4, 5, 6, 7, or 8.
- The number of ways to get each score:
- 2: (1,1) — 1 way
- 3: (1,2), (2,1) — 2 ways
- 4: (1,3), (2,2), (3,1) — (1,3) occurs twice, (2,2) once, (3,1) once — 4 ways
- 5: (2,3), (3,2), (4,1) — (2,3) occurs twice, (3,2) once, (4,1) once — 4 ways
- 6: (3,3), (4,2), (5,1) — (3,3) occurs twice, (4,2) once, (5,1) once — 4 ways
- 7: (4,3), (5,2), (6,1) — (4,3) occurs twice, (5,2) once, (6,1) once — 4 ways
- 8: (5,3), (6,2) — (5,3) occurs twice, (6,2) once — 3 ways
- Total ways: 1 + 2 + 4 + 4 + 4 + 4 + 3 = 22 ways.
- Total possible outcomes: 6 (red) * 6 (blue) = 36.
- Probability = 22/36 = 11/18.
4) A fair dice and a coin are thrown together. Using a sample space diagram:
a. Work out how many possible outcomes there are.
- The dice has 6 outcomes (1 to 6).
- The coin has 2 outcomes (Heads, Tails).
- Total outcomes: 6 * 2 = 12.
b. Work out the probability of getting an even number and a tails.
- Even numbers on dice: 2, 4, 6 — 3 outcomes.
- Tails: 1 outcome.
- Favorable outcomes: (2,T), (4,T), (6,T) — 3 outcomes.
- Probability = 3/12 = 1/4.
5) The royal cards of clubs and the royal cards of diamonds are randomly put into two piles. A card is taken from each pile, work out the probability that:
a. Both cards will be a Queen.
- There are 4 royal cards: King, Queen, Jack, Ace for each suit.
- So, 2 queens (one from clubs, one from diamonds).
- The probability that the first card is a Queen from clubs is 1/4.
- The probability that the second card is a Queen from diamonds is 1/4.
- Since the draws are independent, the probability that both are Queens is (1/4) * (1/4) = 1/16.
b. At least one card will be a Jack.
- The probability that the first card is not a Jack is 3/4.
- The probability that the second card is not a Jack is 3/4.
- The probability that neither is a Jack is (3/4) * (3/4) = 9/16.
- The probability that at least one is a Jack is 1 - 9/16 = 7/16.
Parent Tip: Review the logic above to help your child master the concept of sample space probability worksheet.
