Understand Compound Event Probability - 7th Grade Math

1. What is $P(\text{sum is }7)$ when rolling two fair six-sided dice?

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

2. What is $P(\text{even then odd})$ when rolling a fair die twice?

   $\frac{9}{36}=\frac{1}{4}$. 3 even × 3 odd = 9 favorable outcomes out of 36.

3. A bag has $3$ red and $2$ blue marbles. Draw one marble. What is $P(\text{blue})$?

   $\frac{2}{5}$. Two blue marbles out of five total marbles.

4. What is the sample space size for spinning a $4$-section spinner twice (ordered outcomes)?

   $16$. 4 first spin × 4 second spin = 16 ordered pairs.

5. Identify the sample space size for rolling one fair six-sided die.

   $6$. A die has 6 faces, so 6 possible outcomes.

6. What is $P(\text{sum }=7)$ when rolling two fair six-sided dice?

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

7. Identify the sample space size for flipping $2$ fair coins.

   $4$. Each coin has 2 outcomes: $2 \times 2 = 4$ total.

8. What is $P(\text{at least one }6)$ when rolling a fair die twice?

   $\frac{11}{36}$. 11 outcomes have at least one 6: all with 6 minus double-counting $(6,6)$.

9. What is the definition of probability using the sample space?

   $P(E)=\frac{\text{favorable outcomes}}{\text{total outcomes in sample space}}$

10. What is a compound event in probability?

    An event made from two or more simple events. Combines multiple simple events like "heads AND tails".

11. What is the sample space (as ordered pairs) for flipping $2$ coins?

    ${(H,H),(H,T),(T,H),(T,T)}$. Lists all combinations of H/T for first and second coin.

12. What does the sample space represent for a probability experiment?

    The set of all possible outcomes. Contains every outcome that could happen in an experiment.

13. What is $P(\text{at least one head})$ when flipping $2$ fair coins?

    $\frac{3}{4}$. 3 outcomes have at least one H: (H,H),(H,T),(T,H) out of 4.

14. What is $P(\text{both numbers are }6)$ when rolling a fair die twice?

    $\frac{1}{36}$. Only (6,6) has both dice showing 6 out of 36.

15. What is the sample space for rolling one standard die?

    ${1,2,3,4,5,6}$

16. What is $P(\text{not a heart})$ when drawing one card from a standard $52$-card deck?

    $\frac{39}{52}$. $52 - 13 = 39$ cards are not hearts.

17. What is $P(\text{red and face card})$ from a standard $52$-card deck?

    $\frac{6}{52}$. Red face cards are $6$ total: $3$ red jacks, queens, and kings.

18. From ${A,B,C}$ and ${1,2}$, what is $P(\text{letter }B\text{ or number }2)$ for a random ordered pair?

    $\frac{4}{6}=\frac{2}{3}$. 4 pairs contain B or 2: (B,1),(B,2),(A,2),(C,2) out of 6.

19. What is the sample space size for two fair coin flips?

    $4$. Two flips give outcomes: $\{HH, HT, TH, TT\}$.

20. What is $P(\text{both heads})$ for two fair coin flips?

    $\frac{1}{4}$. Only one outcome $HH$ out of four possible: $\{HH, HT, TH, TT\}$.

21. What is $P(\text{first is }H\text{ and second is }T)$ when flipping $2$ fair coins?

    $\frac{1}{4}$. Only (H,T) matches this specific order out of 4.

22. What is the sample space size for flipping a fair coin and rolling a fair die?

    $12$. Coin has $2$ outcomes, die has $6$, so $2 \times 6 = 12$ total.

23. A bag has $3$ red and $2$ blue marbles. Two are drawn with replacement. What is $P(\text{same color})$?

    $\left(\frac{3}{5}\right)^2+\left(\frac{2}{5}\right)^2=\frac{13}{25}$. P(RR) + P(BB) = $(\frac{3}{5})^2 + (\frac{2}{5})^2$.

24. What is the sample space of an experiment?

    The set of all possible outcomes. Sample space contains every outcome that could occur.

25. A bag has $3$ red and $2$ blue marbles. Two are drawn with replacement. What is $P(R\text{ then }B)$?

    $\frac{3}{5}\cdot\frac{2}{5}=\frac{6}{25}$. With replacement: P(R)×P(B) = $\frac{3}{5} \times \frac{2}{5}$.

26. A spinner has $4$ equal sections labeled $A,B,C,D$. What is $P(A\text{ or }B)$?

    $\frac{2}{4}$. Two favorable sections out of four equal sections.

27. What is the probability of an event written as a fraction of the sample space?

    $P(E)=\frac{\text{number of favorable outcomes}}{\text{total outcomes in the sample space}}$. Favorable outcomes divided by total possible outcomes gives probability.

28. What is $P(\text{even and }\gt 3)$ when rolling one fair six-sided die?

    $\frac{2}{6}$. Even numbers $>3$ are $\{4, 6\}$, so $2$ favorable outcomes.

29. A bag has $3$ red and $2$ blue marbles. Draw one marble. What is $P(\text{red})$?

    $\frac{3}{5}$. Three red marbles out of five total marbles.

30. A class has $12$ students: $7$ wear glasses and $5$ do not. Two students are chosen without replacement. What is $P(G\text{ then }G)$?

    $\frac{7}{12}\cdot\frac{6}{11}=\frac{7}{22}$. Without replacement: $\frac{7}{12}$ for first, then $\frac{6}{11}$ for second.

31. Identify the probability: pick a number from $\{1,2,3,4,5\}$, $P(\text{odd or }5)$.

    $\frac{3}{5}$. Odd numbers are $\{1,3,5\}$, which includes $5$, so $3$ favorable outcomes.

32. What is $P(\text{at least one head})$ for two fair coin flips?

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

33. What is $P(\text{sum }\le 3)$ when rolling two fair six-sided dice?

    $\frac{3}{36}$. Only $(1,1), (1,2), (2,1)$ give sums $\le 3$ out of $36$ outcomes.

34. What is $P(\text{red or face card})$ from a standard $52$-card deck?

    $\frac{32}{52}$. $26$ red cards plus $12$ face cards minus $6$ red face cards.

35. What is a compound event in probability?

    An event made of two or more simple events. Combines multiple simple events like "heads AND tails" or "red OR blue".

36. What is $P(\text{both heads})$ when flipping $2$ fair coins?

    $\frac{1}{4}$. Only (H,H) gives both heads out of 4 outcomes.

37. Which sample space size is correct for rolling one fair six-sided die once?

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

38. A letter is chosen from ${A,B,C}$ and a number from ${1,2}$. What is the sample space size for ordered pairs?

    $3\cdot 2=6$. 3 letters × 2 numbers = 6 ordered pairs.

39. What is $P(\text{exactly one head})$ for two fair coin flips?

    $\frac{2}{4}$. Two outcomes have exactly one head: $HT$ and $TH$.

40. What is the definition of probability using a sample space?

    $P(E)=\frac{\text{number of outcomes in }E}{\text{total outcomes in sample space}}$. Probability equals favorable outcomes divided by total outcomes.

41. What does it mean for outcomes to be equally likely?

    Each outcome has the same probability. All outcomes occur with equal chance.

42. A spinner has $4$ equal sections labeled $1$ to $4$. It is spun twice. What is $P(\text{sum }=5)$?

    $\frac{4}{16}=\frac{1}{4}$. 4 ways to sum to 5: (1,4),(2,3),(3,2),(4,1) out of 16.

43. Identify the correct probability: choosing a heart from a standard $52$-card deck.

    $\frac{1}{4}$. There are $13$ hearts in $52$ cards, so $\frac{13}{52} = \frac{1}{4}$.

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