Develop Uniform Probability Models - 7th Grade Math

1. What is the sample space size for flipping one fair coin?

   $2$ outcomes. A coin has two sides: heads and tails.

2. Identify the probability of rolling a number greater than $4$ on a fair six-sided die.

   $\frac{1}{3}$. Two numbers (5, 6) are greater than 4 out of 6 total.

3. What is the sample space for one fair coin toss?

   ${H,T}$. A coin can land on heads (H) or tails (T).

4. What is the formula for probability in a uniform model using favorable and total outcomes?

   $P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$. Count favorable outcomes and divide by total possible outcomes.

5. Identify the probability that the sum is $7$ when rolling two fair six-sided dice.

   $\frac{1}{6}$. Six ways to get sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

6. Identify the probability of getting two tails in two fair coin tosses.

   $\frac{1}{4}$. Only TT gives two tails out of four equally likely outcomes.

7. Find $P(\text{choose red})$ from $5$ equally likely colors with $2$ red colors.

   $\frac{2}{5}$. $2$ red colors out of $5$ total colors.

8. Identify the probability of drawing an ace from a standard $52$-card deck in a uniform model.

   $\frac{1}{13}$. 4 aces in the deck out of 52 total cards.

9. Identify the probability of selecting a prime number from $\{1,2,3,4,5\}$ in a uniform model.

   $\frac{3}{5}$. Three prime numbers (2, 3, 5) out of 5 total numbers.

10. Identify the probability of rolling a $4$ on a fair six-sided die.

    $\frac{1}{6}$. One favorable outcome (rolling 4) out of 6 equally likely outcomes.

11. Find $P(\text{flip heads})$ for a fair coin.

    $\frac{1}{2}$. Fair coin means heads and tails are equally likely.

12. What is the sample space size for rolling one fair six-sided die?

    $6$ outcomes. A standard die has faces numbered $1$ through $6$.

13. Identify the probability of drawing a face card (J, Q, or K) from a $52$-card deck.

    $\frac{3}{13}$. 12 face cards (3 per suit $\times$ 4 suits) out of 52 total.

14. Identify the probability of rolling an even number on a fair six-sided die.

    $\frac{1}{2}$. Three even numbers (2, 4, 6) out of 6 total outcomes.

15. What is the formula for $P(A)$ in a uniform model using favorable and total outcomes?

    $P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$. Count favorable outcomes divided by total possible outcomes.

16. Find $P(\text{flip tails})$ for a fair coin.

    $\frac{1}{2}$. Fair coin means heads and tails are equally likely.

17. What formula gives probability in a uniform model using favorable and total outcomes?

    $P(A)=\frac{\text{favorable outcomes}}{\text{total outcomes}}$. Count favorable outcomes and divide by total possible outcomes.

18. What is the probability of drawing a red card from a standard $52$-card deck in a uniform model?

    $\frac{1}{2}$. 26 red cards (hearts and diamonds) out of 52 total cards.

19. What is the sample space size for flipping $2$ fair coins?

    $4$ outcomes. Possible outcomes: HH, HT, TH, TT.

20. What is the probability of one outcome in a uniform model with $n$ equally likely outcomes?

    $P(\text{one outcome})=\frac{1}{n}$. Each outcome has equal probability in a uniform model.

21. Find $P(\text{exactly one head})$ when flipping $2$ fair coins.

    $\frac{2}{4}=\frac{1}{2}$. HT and TH each have exactly one head.

22. Identify the probability of getting exactly one head in two fair coin tosses.

    $\frac{1}{2}$. HT and TH are the two ways to get exactly one head out of four outcomes.

23. What is the probability of landing on an odd number on a fair spinner labeled $1$ through $10$?

    $\frac{1}{2}$. 5 odd numbers (1, 3, 5, 7, 9) out of 10 total numbers.

24. What is the sample space for one flip of a fair coin?

    ${H,T}$. A coin has two possible outcomes: heads or tails.

25. Identify the correct probability for a certain event in any probability model.

    $1$. A certain event always occurs.

26. Identify the probability of rolling doubles with two fair six-sided dice.

    $\frac{1}{6}$. Six doubles: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) out of 36.

27. What is the sample space for two fair coin tosses written as ordered pairs?

    ${HH,HT,TH,TT}$. List all possible ordered outcomes for two coin flips.

28. What is the probability of selecting a vowel from the letters in $\text{MATH}$, assuming each letter is equally likely?

    $\frac{1}{4}$. One vowel (A) out of 4 letters total.

29. Find $P(\text{not }A)$ if a uniform model gives $P(A)=\frac{3}{8}$.

    $P(\text{not }A)=\frac{5}{8}$. $P(\text{not }A) = 1 - P(A) = 1 - \frac{3}{8} = \frac{5}{8}$.

30. Find $P(\text{roll an even number})$ when rolling a fair six-sided die.

    $\frac{3}{6}=\frac{1}{2}$. Even numbers are $2, 4, 6$, so $3$ out of $6$ outcomes.

31. Find $P(\text{roll a number greater than }4)$ on a fair six-sided die.

    $\frac{2}{6}=\frac{1}{3}$. Only $5$ and $6$ are greater than $4$.

32. Identify the probability of drawing an ace from a standard $52$-card deck.

    $\frac{1}{13}$. 4 aces (one per suit) out of 52 total cards.

33. Identify the probability of drawing a heart from a standard $52$-card deck.

    $\frac{1}{4}$. 13 hearts out of 52 total cards in a standard deck.

34. Identify the probability that the sum is $2$ when rolling two fair six-sided dice.

    $\frac{1}{36}$. Only (1,1) gives sum 2 out of 36 possible outcomes.

35. Identify the probability of getting at least one head in two fair coin tosses.

    $\frac{3}{4}$. Three outcomes (HH, HT, TH) have at least one head out of four total.

36. Find $P(\text{not }A)$ if $P(A)=\frac{2}{5}$ in a uniform probability setting.

    $\frac{3}{5}$. The complement probability is $1 - P(A) = 1 - \frac{2}{5}$.

37. Find $P(\text{choose a vowel})$ when selecting one letter from $\text{MATH}$ uniformly.

    $\frac{1}{4}$. Only A is a vowel among the $4$ letters.

38. What is the probability of one specific outcome in a uniform model with $n$ outcomes?

    $\frac{1}{n}$. Each outcome has equal probability, so divide 1 by the number of outcomes.

39. What is the sample space size for choosing one letter uniformly from the word $\text{MATH}$?

    $4$ outcomes. MATH has $4$ distinct letters: M, A, T, H.

40. Identify the correct probability for an impossible event in any probability model.

    $0$. An impossible event never occurs.

41. What is the sample space size for rolling two fair six-sided dice (ordered outcomes)?

    $36$. Each die has 6 outcomes, so $6 \times 6 = 36$ ordered pairs.

42. Identify the error: In a uniform model with $10$ outcomes, a student says $P(\text{one outcome})=\frac{1}{9}$.

    Correct probability is $\frac{1}{10}$. With 10 outcomes, each has probability $\frac{1}{10}$, not $\frac{1}{9}$.

43. Find $P(A)$ if a uniform model has $20$ outcomes and event $A$ has $7$ favorable outcomes.

    $\frac{7}{20}$. Apply the formula with $7$ favorable and $20$ total outcomes.

44. Find $P(\text{sum }=2)$ when rolling $2$ fair six-sided dice.

    $\frac{1}{36}$. Only one way to get sum $2$: rolling $(1,1)$.

45. Identify the probability of drawing a red card from a standard $52$-card deck.

    $\frac{1}{2}$. 26 red cards (13 hearts + 13 diamonds) out of 52 total.

46. Identify the probability of choosing a multiple of $3$ from $\{1,2,3,4,5,6,7,8,9\}$ uniformly.

    $\frac{1}{3}$. Three multiples of 3 (3, 6, 9) out of 9 total numbers.

47. What is the probability of drawing a heart from a standard $52$-card deck in a uniform model?

    $\frac{1}{4}$. 13 hearts in the deck out of 52 total cards.

48. What is the sample space for one roll of a fair six-sided die?

    ${1,2,3,4,5,6}$. A standard die has six faces numbered 1 through 6.

49. Find $P(\text{sum }=7)$ when rolling $2$ fair six-sided dice.

    $\frac{6}{36}=\frac{1}{6}$. Six ways to get sum $7$: $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$.

50. What is the probability of one specific outcome if a uniform sample space has $n$ outcomes?

    $\frac{1}{n}$. Each outcome has equal probability, so divide 1 by the number of outcomes.

51. What is the sample space size for rolling $2$ fair six-sided dice?

    $36$ outcomes. Each die has $6$ faces, so $6 \times 6 = 36$ combinations.

52. Identify the probability of spinning blue on a spinner with $8$ equal sections if $3$ sections are blue.

    $\frac{3}{8}$. 3 blue sections out of 8 equal sections total.

53. Identify the probability of getting at least one head in two fair coin flips.

    $\frac{3}{4}$. Three favorable outcomes (HH, HT, TH) out of 4 total outcomes.

54. What is a uniform probability model?

    A model where all outcomes in the sample space have equal probability. Each outcome has the same chance of occurring.

55. Identify the probability of getting heads on a fair coin flip.

    $\frac{1}{2}$. One favorable outcome (H) out of 2 equally likely outcomes.

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