Probability - SAT Math

1. What is the probability of rolling a sum of 7 with two dice?

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

2. What is the probability of rolling a 3 on a fair 6-sided die?

   Probability = $\frac{1}{6}$. One favorable outcome (rolling 3) out of six possible outcomes.

3. What is the probability of drawing a diamond from a deck of 52 cards?

   $\frac{1}{4}$. 13 diamonds out of 52 total cards.

4. If $P(A) = 0.3$ and $P(B) = 0.4$, find $P(A \text{ or } B)$ assuming independence.

   $0.58$. Using addition rule: $0.3 + 0.4 - (0.3 \times 0.4) = 0.58$.

5. State the formula for the probability of the complement of an event.

   P(A') = 1 - P(A). Complement probability equals 1 minus the original event's probability.

6. If $P(A) = 0.5$, what is $P(A')$?

   $0.5$. Complement of A: $1 - 0.5 = 0.5$.

7. What is the probability of drawing an ace or a king from a deck?

   $\frac{2}{13}$. 8 favorable cards (4 aces + 4 kings) out of 52 total.

8. What is the sum of probabilities for all possible outcomes of an event?

   1. All possible outcomes together form the complete sample space.

9. What is the probability of rolling a prime number on a 6-sided die?

   $\frac{1}{2}$. Prime numbers on die: 2, 3, 5 (three out of six).

10. What is the probability of drawing a heart from a standard deck of 52 cards?

    $\frac{1}{4}$. 13 hearts out of 52 total cards simplifies to $\frac{1}{4}$.

11. Identify the probability of rolling an even number on a fair 6-sided die.

    Probability = $\frac{3}{6}$ or $\frac{1}{2}$. Three even numbers (2, 4, 6) out of six possible outcomes.

12. What is the probability of flipping two heads on two fair coins?

    $\frac{1}{4}$. Multiply independent probabilities: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.

13. State the formula for conditional probability $P(A|B)$.

    $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$. Probability of A given B has occurred.

14. Define independent events.

    Events where the occurrence of one does not affect the other. One event's outcome doesn't change the other's probability.

15. What is the probability of drawing a red ace from a deck of cards?

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

16. What is the probability of drawing a black card from a standard deck?

    $\frac{1}{2}$. 26 black cards (spades and clubs) out of 52 total.

17. Define mutually exclusive events.

    Events that cannot occur simultaneously. If one occurs, the other cannot happen at the same time.

18. Define exhaustive events.

    A set of events that cover all possible outcomes. Events whose probabilities sum to 1 (complete sample space).

19. What is the probability of at least one head in two coin flips?

    $\frac{3}{4}$. Complement of all tails: $1 - \frac{1}{4} = \frac{3}{4}$.

20. If two events A and B are mutually exclusive, what is $P(A \text{ and } B)$?

    $P(A \text{ and } B) = 0$. Mutually exclusive events cannot happen together.

21. State the formula for the probability of independent events A and B both occurring.

    $P(A \text{ and } B) = P(A) \times P(B)$. For independent events, multiply individual probabilities.

22. What is the probability of rolling an even number on a 6-sided die?

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

23. What is the probability formula for a single event occurring?

    Probability = (Number of favorable outcomes) / (Total number of possible outcomes). This is the fundamental definition of probability in statistics.

24. Identify the probability of drawing a heart from a standard deck of cards.

    1/4. 13 hearts out of 52 total cards in a standard deck.

25. What is the probability of drawing a queen from a shuffled deck of 52 cards?

    $\frac{1}{13}$. 4 queens out of 52 cards simplifies to $\frac{1}{13}$.

26. What is the probability of not drawing a face card from a deck of cards?

    $\frac{10}{13}$. 40 non-face cards out of 52 total cards.

27. What is the probability of rolling a 4 on a fair 6-sided die?

    $\frac{1}{6}$. One favorable outcome out of six possible outcomes.

28. What is the probability of flipping a coin and getting heads?

    Probability = $\frac{1}{2}$. One favorable outcome (heads) out of two possible outcomes.

29. What is the probability of drawing a red card after drawing a black card without replacement?

    $\frac{26}{51}$. After removing one black card, 26 red remain out of 51 total.

30. Identify the probability of drawing an ace from a standard deck of 52 cards.

    Probability = $\frac{4}{52}$ or $\frac{1}{13}$. Four aces in a standard deck of 52 cards.

31. What is the probability of drawing a card that is either a club or a face card?

    $\frac{5}{13}$. 13 clubs + 12 face cards - 3 club face cards = 22 favorable.

32. Identify the probability of drawing a heart from a standard deck of 52 cards.

    Probability = $\frac{13}{52}$ or $\frac{1}{4}$. Thirteen hearts in a standard deck of 52 cards.

33. State the addition rule for probability.

    $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. General formula accounting for overlap between events.

34. What is $P(A \text{ or } B)$ if $P(A) = 0.3$, $P(B) = 0.4$, and $P(A \text{ and } B) = 0.2$?

    $0.5$. Apply addition rule: $0.3 + 0.4 - 0.2 = 0.5$.

35. What is the probability of getting a sum of 7 when rolling two fair 6-sided dice?

    Probability = $\frac{6}{36}$ or $\frac{1}{6}$. Six ways to make 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) out of 36 total.

36. State the addition rule for probability of mutually exclusive events.

    P(A or B) = P(A) + P(B). For mutually exclusive events, probabilities add since they cannot occur together.

37. State the formula for calculating probability.

    $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. Basic probability definition using favorable over total outcomes.

38. What is the probability of getting at least one 6 when rolling two dice?

    $\frac{11}{36}$. Use complement: $1 - P(\text{no 6s}) = 1 - \frac{25}{36} = \frac{11}{36}$.

39. What is the probability of not rolling a 3 on a fair 6-sided die?

    $\frac{5}{6}$. Complement of rolling a 3: $1 - \frac{1}{6} = \frac{5}{6}$.

40. What is the probability of getting two tails in three coin flips?

    $\frac{3}{8}$. Count outcomes with exactly two tails: TTH, THT, HTT.

41. State the multiplication rule for independent events.

    $P(A \text{ and } B) = P(A) \times P(B)$. For independent events, probability of both occurring.

42. Define complementary events.

    Two events are complementary if one event is the complement of the other. One event is exactly the opposite of the other.

43. What is the probability of drawing a spade or a face card from a deck?

    $\frac{11}{26}$. 13 spades + 12 face cards - 3 overlap = 22 favorable outcomes.

44. Identify the probability of rolling a 4 on a standard 6-sided die.

    1/6. One specific outcome (rolling 4) out of six equally likely outcomes.

45. What is the probability of drawing a red card from a standard deck of 52 cards?

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

46. What is the probability of drawing a red queen or a black king from a deck?

    $\frac{1}{13}$. 2 red queens + 2 black kings = 4 favorable cards out of 52.

47. What is the probability of rolling a total of 4 with two dice?

    $\frac{1}{12}$. 3 ways to roll 4: (1,3), (2,2), (3,1) out of 36 possibilities.

48. What is $P(A \text{ and } B)$ for independent events A and B?

    $P(A \text{ and } B) = P(A) \times P(B)$. For independent events, multiply their individual probabilities.

49. What is the probability of drawing two aces in a row from a deck without replacement?

    $\frac{1}{221}$. First ace: $\frac{4}{52}$, second ace: $\frac{3}{51}$, multiply together.

50. What is $P(A \text{ and } B)$ if $P(A) = 0.3$ and $P(B) = 0.2$ assuming independence?

    $0.06$. Multiply independent probabilities: $0.3 \times 0.2 = 0.06$.

51. What is the probability of drawing a king or a heart from a deck of cards?

    $\frac{4}{13}$. 4 kings + 13 hearts - 1 king of hearts = 16 favorable.

52. State the formula for the probability of the complement of event A.

    $P(A') = 1 - P(A)$. The complement probability equals 1 minus the event probability.

53. Define dependent events.

    Events where the occurrence of one affects the probability of the other. One event's outcome changes the probability of the other.

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